Thanks for the reply. I upgraded from 5.803 to 5.805 (no difference). I learned a bit more about monitoring this, and (much to my surprise), the first program above didn't cause any problems at all (the connection closes almost immediately). The last part of the tcpdump: 
14:41:12.600732 IP (tos 0x0, ttl  49, id 47053, offset 0, flags [DF], 
+proto: TCP (6), length: 1460) host.example.com.http > system.localdom
+ain.loc.53567: . 85889:87297(1408) ack 174 win 54 <nop,nop,timestamp 
+1545425010 745586731>
14:41:12.706665 IP (tos 0x0, ttl  49, id 47054, offset 0, flags [DF], 
+proto: TCP (6), length: 1460) host.example.com.http > system.localdom
+ain.loc.53567: . 87297:88705(1408) ack 174 win 54 <nop,nop,timestamp 
+1545425010 745586731>
14:41:12.706694 IP (tos 0x0, ttl  64, id 25068, offset 0, flags [DF], 
+proto: TCP (6), length: 52) system.localdomain.loc.53567 > host.examp
+le.com.http: ., cksum 0xf112 (correct), ack 88705 win 16022 <nop,nop,
+timestamp 745589918 1545425010>
14:41:12.812632 IP (tos 0x0, ttl  49, id 47055, offset 0, flags [DF], 
+proto: TCP (6), length: 1460) host.example.com.http > system.localdom
+ain.loc.53567: . 88705:90113(1408) ack 174 win 54 <nop,nop,timestamp 
+1545425010 745586731>
14:41:12.842084 IP (tos 0x0, ttl  49, id 47056, offset 0, flags [DF], 
+proto: TCP (6), length: 355) host.example.com.http > system.localdoma
+in.loc.53567: FP 90113:90416(303) ack 174 win 54 <nop,nop,timestamp 1
+545425010 745586731>
14:41:12.842123 IP (tos 0x0, ttl  64, id 25069, offset 0, flags [DF], 
+proto: TCP (6), length: 52) system.localdomain.loc.53567 > host.examp
+le.com.http: ., cksum 0xe9db (correct), ack 90417 win 16022 <nop,nop,
+timestamp 745590053 1545425010>
14:41:12.844363 IP (tos 0x0, ttl  64, id 25070, offset 0, flags [DF], 
+proto: TCP (6), length: 52) system.localdomain.loc.53567 > host.examp
+le.com.http: F, cksum 0xe9d7 (correct), 174:174(0) ack 90417 win 1602
+2 <nop,nop,timestamp 745590056 1545425010>
14:41:13.071450 IP (tos 0x0, ttl  49, id 0, offset 0, flags [DF], prot
+o: TCP (6), length: 52) host.example.com.http > system.localdomain.lo
+c.53567: ., cksum 0x24f8 (correct), ack 175 win 54 <nop,nop,timestamp
+ 1545425842 745590056>

[download]
The second program above (maybe because of the POST method) causes all of the problems, with each connection staying open for several minutes: 
16:34:08.538882 IP (tos 0x0, ttl  49, id 57175, offset 0, flags [DF], 
+proto: TCP (6), length: 1460) host.example.com.http > system.localdom
+ain.loc.35789: . 9857:11265(1408) ack 291 win 54 <nop,nop,timestamp 1
+547119394 752366263>
16:34:08.538899 IP (tos 0x0, ttl  64, id 63133, offset 0, flags [DF], 
+proto: TCP (6), length: 52) system.localdomain.loc.35789 > host.examp
+le.com.http: ., cksum 0x00d5 (correct), ack 11265 win 7252 <nop,nop,t
+imestamp 752366807 1547119394>
16:34:08.644817 IP (tos 0x0, ttl  49, id 57176, offset 0, flags [DF], 
+proto: TCP (6), length: 1460) host.example.com.http > system.localdom
+ain.loc.35789: P 11265:12673(1408) ack 291 win 54 <nop,nop,timestamp 
+1547119394 752366263>
16:34:08.644831 IP (tos 0x0, ttl  64, id 63134, offset 0, flags [DF], 
+proto: TCP (6), length: 52) system.localdomain.loc.35789 > host.examp
+le.com.http: ., cksum 0xf816 (correct), ack 12673 win 7976 <nop,nop,t
+imestamp 752366913 1547119394>
16:34:08.664134 IP (tos 0x0, ttl  49, id 57177, offset 0, flags [DF], 
+proto: TCP (6), length: 238) host.example.com.http > system.localdoma
+in.loc.35789: P 12673:12859(186) ack 291 win 54 <nop,nop,timestamp 15
+47119423 752366376>
16:34:08.664158 IP (tos 0x0, ttl  64, id 63135, offset 0, flags [DF], 
+proto: TCP (6), length: 52) system.localdomain.loc.35789 > host.examp
+le.com.http: ., cksum 0xf46c (correct), ack 12859 win 8680 <nop,nop,t
+imestamp 752366932 1547119423>
16:34:08.674181 IP (tos 0x0, ttl  49, id 57178, offset 0, flags [DF], 
+proto: TCP (6), length: 73) host.example.com.http > system.localdomai
+n.loc.35789: P, cksum 0x015b (correct), 12859:12880(21) ack 291 win 5
+4 <nop,nop,timestamp 1547119431 752366376>
16:34:08.674194 IP (tos 0x0, ttl  64, id 63136, offset 0, flags [DF], 
+proto: TCP (6), length: 52) system.localdomain.loc.35789 > host.examp
+le.com.http: ., cksum 0xf445 (correct), ack 12880 win 8680 <nop,nop,t
+imestamp 752366942 1547119431>
16:34:08.674692 IP (tos 0x0, ttl  64, id 63137, offset 0, flags [DF], 
+proto: TCP (6), length: 52) system.localdomain.loc.35789 > host.examp
+le.com.http: F, cksum 0xf443 (correct), 291:291(0) ack 12880 win 8680
+ <nop,nop,timestamp 752366943 1547119431>
16:34:08.680820 IP (tos 0x0, ttl  49, id 57179, offset 0, flags [DF], 
+proto: TCP (6), length: 52) host.example.com.http > system.localdomai
+n.loc.35789: F, cksum 0x182d (correct), 12880:12880(0) ack 291 win 54
+ <nop,nop,timestamp 1547119431 752366376>
16:34:08.680858 IP (tos 0x0, ttl  64, id 63138, offset 0, flags [DF], 
+proto: TCP (6), length: 52) system.localdomain.loc.35789 > host.examp
+le.com.http: ., cksum 0xf43c (correct), ack 12881 win 8680 <nop,nop,t
+imestamp 752366949 1547119431>
16:34:09.828456 IP (tos 0x0, ttl  64, id 63139, offset 0, flags [DF], 
+proto: TCP (6), length: 52) system.localdomain.loc.35789 > host.examp
+le.com.http: F, cksum 0xefc0 (correct), 291:291(0) ack 12881 win 8680
+ <nop,nop,timestamp 752368097 1547119431>
16:34:12.136097 IP (tos 0x0, ttl  64, id 63140, offset 0, flags [DF], 
+proto: TCP (6), length: 52) system.localdomain.loc.35789 > host.examp
+le.com.http: F, cksum 0xe6bc (correct), 291:291(0) ack 12881 win 8680
+ <nop,nop,timestamp 752370405 1547119431>
16:34:16.751393 IP (tos 0x0, ttl  64, id 63141, offset 0, flags [DF], 
+proto: TCP (6), length: 52) system.localdomain.loc.35789 > host.examp
+le.com.http: F, cksum 0xd4b4 (correct), 291:291(0) ack 12881 win 8680
+ <nop,nop,timestamp 752375021 1547119431>
16:34:25.981983 IP (tos 0x0, ttl  64, id 63142, offset 0, flags [DF], 
+proto: TCP (6), length: 52) system.localdomain.loc.35789 > host.examp
+le.com.http: F, cksum 0xb0a4 (correct), 291:291(0) ack 12881 win 8680
+ <nop,nop,timestamp 752384253 1547119431>

[download]
I can't really decipher everything going on there. If you have any additional insights, please share them. Thank you again for your help.

In reply to Re^2: Closing the connection by Anonymous Monk
in thread Closing the connection by Anonymous Monk

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.