If I set the relative_url_root in config/environment.rb using
Redmine::Utils::relative_url_root = "/redmine"
or
ActionController::Base.relative_url_root = "/redmine"
or with an ENV of RAILS_RELATIVE_URL_ROOT=/redmine

the script and link elements correct have the /redmine/.... in the
src and href fields but the href in the anchor (a) elements like
for menus (the only ones I have looks at yet) do not have the
leading '/redmine'

I am running a brand new install of redmine on fedora 29 with
postgersql 10.7
and a info page of
Information

Redmine 4.0.3.stable
Default administrator account changed (checked)
Attachments directory writable (checked)
Plugin assets directory writable (./public/plugin_assets) (checked)
RMagick available (optional) (checked)
ImageMagick convert available (optional) (checked)

Environment:
Redmine version 4.0.3.stable
Ruby version 2.5.5-p157 (2019-03-15) [x86_64-linux]
Rails version 5.2.2.1
Environment production
Database adapter PostgreSQL
Mailer queue ActiveJob::QueueAdapters::AsyncAdapter
Mailer delivery sendmail
SCM:
Subversion 1.11.1
Mercurial 4.5.3
Bazaar 2.7.0
Git 2.20.1
Filesystem
Redmine plugins:
no plugin installed