Put computer to sleep and reawaken and Lichess Local Engine doesn't respond to Analysis, even if running

1. Open Lichess Local Engine
2. Open lichess analysis and configure it to use external engines
3. Run the engine and see it run well
4. Put computer to sleep
5. Reawaken it
6. Run the engine in the analysis page and it doesn't respond

Windows 11

Reproduce the Windows 11 sequence: run the Lichess Local Engine, use it from lichess analysis, suspend and resume the computer, then try analysis again. Start by tracing how the local engine process and analysis connection are managed across system sleep; done means the engine responds to analysis after wake without requiring a restart.

Written by the indexing model from the issue text.