Duplicate tray icon appears after closing window when "Keep Bitwarden running in the background" is enabled on KDE PLasma 6.7 (Flatpak)

### Steps To Reproduce

1. Enable **Keep Bitwarden running in the background** in app settings
2. Start Bitwarden
3. Observe a single tray icon in Systemtray
4. Close main Window (via Window controll)

### Expected Result

Existing tray icon should continue to be used instead of creating a second one

### Actual Result

A second BitWarden tray icon immediately appears.
The Plasma System Tray configuration now shows two seperate Bitwarden entries (with the same name but slightly different icons (icon size))

### Screenshots or Videos

### Additional Context

- Only one Bitwarden application instance is running
- No duplicate Plasma Autostart entries exist
- Disabling **Keep BitWarden running in the background** removes all Bitwarden tray icons
- Ferdium (Electron Flatpak) behaces correctly on the same system and only ever shows a single tray icon
- the issue is reproducible every time

Environment:
_cat /etc/os-release_
PRETTY_NAME="KDE neon User Edition"
NAME="KDE neon"
VERSION_ID="24.04"
VERSION="User Edition"
VERSION_CODENAME=noble

_kinfo_
Operating System: KDE neon User Edition
KDE Plasma Version: 6.7.3
KDE Frameworks Version: 6.28.0
Qt Version: 6.11.1
Kernel Version: 7.0.0-28-generic (64-bit)
Graphics Platform: Wayland
Processors: 4 × Intel® Core™ i5-6200U CPU @ 2.30GHz
Memory: 17 GB of RAM (16.7 GB usable)
Graphics Processor: Intel® HD Graphics 520

### Operating System

Linux

### Operating System Version

KDE Neon 24.04 User Edition, Plasmashell Version 6.7.3

### Installation method

Flatpak

### Build Version

2026.7.0

### Issue Tracking Info

- [x] I understand that work is tracked outside of GitHub. A PR will be linked to this issue should one be opened to address it, but Bitwarden doesn't use fields like "assigned", "milestone", or "project" to track progress.

Reproduce the issue in the Flatpak build on KDE Plasma with background running enabled, then trace the desktop tray lifecycle used when the main window closes. Verify how the existing tray icon is retained or recreated. Done means closing the window leaves exactly one Bitwarden tray entry while disabling background running still removes the icon.

Written by the indexing model from the issue text.