feat(code/app): Add persistence to the example app

[romac]

Closes: #740

---

PR author checklist

For all contributors

• Reference GitHub issue
• Ensure PR title follows the conventional commits spec

For external contributors

• Maintainers of Malachite are allowed to push changes to the branch

[codecov]

Codecov Report

Attention: Patch coverage is 40.00000% with 3 lines in your changes missing coverage. Please review.

Project coverage is 79.28%. Comparing base (b3a7a67) to head (6c05791).
Report is 1 commits behind head on main.

✅ All tests successful. No failed tests found.

Files with missing lines	Patch %	Lines
code/crates/core-types/src/proposal.rs	0.00%	3 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #746      +/-   ##
==========================================
- Coverage   79.50%   79.28%   -0.21%     
==========================================
  Files         104      104              
  Lines        9023     9026       +3     
==========================================
- Hits         7173     7156      -17     
- Misses       1850     1870      +20     

Flag	Coverage Δ
integration	78.99% <40.00%> (-0.50%)	⬇️
mbt	20.07% <0.00%> (?)

Flags with carried forward coverage won't be shown. Click here to find out more.

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.