Custom cop instantiated multiple times when support_multiple_source? returns true

[karouf]

I have a use case where it would be handy to have a custom cop only instantiated once for the whole Rubocop run. After reading the v1 upgrade notes and the matching API doc, I got the idea that implementing on my custom cop a #support_mulitple_source? class method that would return true should do the trick.
Unfortunately, it seems I either missed (or misunderstood) the docs or there is a bug with this feature.
It seems to have been introduced with #7868 which main point wasn't that specific feature but autocorrection API. Which could explain if there was an issue with it. Also I'm not sure it's really used that much.
I extracted a small git repo to make it easier to reproduce the issue: https://github.com/karouf/rubocop_support_multiple_source

Expected behavior

The cop is instantiated only once, therefore the @test instance variable is set when the first file is scanned and its value only displayed once.

$ bundle exec rubocop --cache false
Inspecting 2 files
@test not assigned yet
.
.

2 files inspected, no offenses detected

Actual behavior

The cop is instanciated for each file that Rubocop scans, and the value of @test is displayed twice.

$ bundle exec rubocop --cache false --debug
For /home/renaud/code/rubocop_multiple_sources: configuration from /home/renaud/code/rubocop_multiple_sources/.rubocop.yml
Default configuration from /home/renaud/.rbenv/versions/3.1.2/lib/ruby/gems/3.1.0/gems/rubocop-1.37.1/config/default.yml
Inspecting 2 files
Scanning /home/renaud/code/rubocop_multiple_sources/file1.rb
@test not assigned yet
.Scanning /home/renaud/code/rubocop_multiple_sources/file2.rb
@test not assigned yet
.

2 files inspected, no offenses detected
Finished in 0.11073834299986629 seconds

Steps to reproduce the problem

git clone https://github.com/karouf/rubocop_support_multiple_source.git
cd rubocop_support_multiple_source
bundle install
bundle exec rubocop

RuboCop version

 $ bundle exec rubocop -V
1.37.1 (using Parser 3.1.2.1, rubocop-ast 1.23.0, running on ruby 3.1.2) [x86_64-linux]