Godot 4.6 on steam fails to close - stuck in "stopping"

### Tested versions

Reproducable in: 4.6 steam version (the latest version as of 31/01/2026)
Didn't reproduce in the last 4.5 version on steam

### System information

Godot v4.6.stable (89cea1439) - Windows 11 (build 26200) - Multi-window, 2 monitors - Vulkan (Forward+) - dedicated NVIDIA GeForce GTX 1060 3GB (NVIDIA; 32.0.15.7283) - Intel(R) Core(TM) i7-9700 CPU @ 3.00GHz (8 threads) - 31.93 GiB memory

Steam Beta Branch: Stable Client
Steam Version: 1769025840
Steam Client Build Date: Wed, Jan 21 7:13 PM UTC -08:00
Steam Web Build Date: Wed, Jan 21 9:41 PM UTC -08:00
Steam API Version: SteamClient023

### Issue description

Since updating to 4.6, whenever I open the launcher from steam and quit, steam still thinks godot is running:

Clicking STOP doesn't work, and gets stuck in STOPPING:

This also happens when I open my project, or even a new one.

### Steps to reproduce

Click PLAY in steam
optionally open a project
close godot

steam will show the game as still running
Godot won't appear in the task manager

### Minimal reproduction project (MRP)

The launcher is enough

Use the Steam-distributed launcher as the minimal reproduction: click PLAY, close Godot, and confirm Steam remains in STOPPING while Godot is absent from the task manager. Trace the launcher’s Steam launch and exit integration, then verify that closing the launcher lets Steam recognize the process as stopped.

Written by the indexing model from the issue text.