Run client test suite in the CI

[paluh]

Pre-submit checklist:

• Branch
  • [ ] Tests are provided (if possible)
  • [x] Commit sequence broadly makes sense
  • [x] Key commits have useful messages
  • [x] Relevant tickets are mentioned in commit messages
  • [x] Formatting, materialized Nix files, PNG optimization, etc. are updated
• PR
  • [x] Self-reviewed the diff
  • [ ] Useful pull request description
  • [ ] Reviewer requested