Clicking on the cog item adds an additional '/jenkins/' to the path when clicking on the cog item causing a 404 error.

Incorrect url = https://jenkins.***.***/jenkins/job/********/configure
Correct url = https://jenkins.***.***/job/********/configure

Administration and Logout/login buttons go to the correct path without the additional '/jenkins/'

Edit: this is particularly nasty. regression introduced in this commit: https://github.com/jenkinsci/blueocean-plugin/commit/d4a615534c8f8a5e94affc1a612038be90b109f9 with a refactoring.
This would never have worked when running on root - it would always fall back to pre-prepending /jenkins, as it didn't detect that the root was set, correctly.

tfennelly proposed fix: https://github.com/jenkinsci/blueocean-plugin/pull/859 - are you able to look at that, do a beta publish dance and adjust that PR to make it testable? it worked locally when I tried it (changes suggested by josh, but there may be a better way). Basically empty string is "falsey" in javascript, so that would never have worked, ever.