aabadcaa
(?^:^((a*)b|a*b?d)*c\2$)
No

aababdcaa
(?^:^((a*)b|a*b?d)*c\2$)
No

aabadcaa
(?^:^(a*b?d|(a*)b)*c\2$)
Yes

aababdcaa
(?^:^(a*b?d|(a*)b)*c\2$)
Yes

This is perl 5, version 38, subversion 0 (v5.38.0) built for x86_64-linux-thread-multi