Try adding:

$/ = undef;
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after your 'use' statements:

#!/usr/bin/perl

use strict;
use warnings;

$/ = undef;

my $s1 = <STDIN>;
print("\$s1= <$s1> \n");

if (<STDIN>) {
  my $s2 = <STDIN>;
  print("\$s2= <$s1> \n")
}
[download]

By not defining (undef) the input record separator (default newline), it will read to EOF.