Icon shown from a specific app is always hidden (position not saved)

### Describe the bug

I use [this plugin](https://github.com/BoD/NyanTrayIntelliJPlugin/) for IntelliJ which shows an icon in the menu bar (full disclosure: I am the author!).

When it is showing, I can drag it to the right of Hidden's `|` icon, so it is visible even when in collapsed mode. ✅

However, when the icon disappears and re-appears later, it goes again to the left of all other icons, and therefore it's hidden again. ❌

I am not sure how Hidden recognizes the owner of an icon to retrieve its saved position? But in this case it doesn't seem to work. For the record the process showing this icon in my case is `java`.

I realize this is a bit "niche" but as I am a developer maybe there is something I can do to help investigate / make this work as intended?

### Steps to reproduce

1. Install and run [IntelliJ Idea](https://www.jetbrains.com/idea/download/#section=mac)
2. Install the [NyanTray plugin](https://github.com/BoD/NyanTrayIntelliJPlugin/)
3. Compile something
4. See the Nyan cat icon appearing on the left of the icons (with the hidden ones)
5. Drag it to the right (with the visible ones)
6. Compile something again
7. See the Nyan cat icon appearing on the left again :(

### Expected behaviour

Hidden should remember where the icon was.

**Screenshots**

After dragging:

After it re-appears:

### Environment

You may write here the specifications like the version of the project, operating system, or hardware if applicable.

**Desktop (please complete the following information):**
- OS: Ventura 13.0.1 (22A400)
- Version: 1.8 (12)