Menu bar items vanish and expand stops working after connecting an external 4K display (v1.10, macOS 26.6.2)

### Description

When I use the MacBook standalone and then connect it to an external 4K display, Hidden Bar loses every menu bar item it manages. The area to the right of the separator stays completely empty, and clicking the expand control does nothing — there is no way to get the icons back from the UI.

Quitting and relaunching Hidden Bar restores everything immediately, so the persisted state is fine; the running instance seems to fail to re-register its status items when the display configuration changes.

### Steps to reproduce

1. Use the MacBook with the built-in display only, with Hidden Bar managing several menu bar items.
2. Connect an external 4K display (mine runs a 2560x1440 HiDPI scaled mode, so the backing store is 5120x2880).
3. Look at the menu bar.

### Expected

Managed icons remain available, and the expand control still toggles the hidden section.

### Actual

All managed icons are gone. Only the system items (battery, Wi-Fi, Control Center, clock) remain, with an empty gap where the managed icons used to be. The expand control does not respond.

### Evidence that this is not the auto-hide timer

I captured the menu bar every 5 seconds for 60 seconds (12 frames) and counted bright pixels in the region between the separator and the battery icon. The value was identical in all 12 frames. With `isAutoHide = 1` and `numberOfSecondForAutoHide = 30`, an auto-hide cycle would have changed that region at least once in that window.

The stored preferences were intact while the bug was visible:

```
"NSStatusItem Preferred Position hiddenbar_expandcollapse" = 441;
"NSStatusItem Preferred Position hiddenbar_separate" = 473;
alwaysHiddenSectionEnabled = 0;
areSeparatorsHidden = 1;
isAutoHide = 1;
isAutoStart = 1;
numberOfSecondForAutoHide = 30;
useFullStatusBarOnExpandEnabled = 0;
```

### Note on a setup that did NOT reproduce it

Previously I used a 2K display with HiDPI forced on, and this problem never occurred there. So forcing a scaled/HiDPI mode by itself does not seem to be enough to trigger it — the 4K panel is what makes the difference in my case.

### Workaround

Quit and relaunch Hidden Bar. The icons come back right away with the same arrangement.

### Environment

- Hidden Bar 1.10 (build 15), installed via Homebrew cask
- macOS 26.6.2, Apple silicon
- External 4K display, 2560x1440 HiDPI scaled mode

### Related

#369 describes icons losing their arranged positions, but does not identify display reconnection as the trigger. #228 and #231 are about notched built-in displays, which is a different situation.

No source files or tests are named. First reproduce the issue by connecting the specified 4K HiDPI display while Hidden Bar is running, then trace how the app handles display configuration changes and status-item registration; done means managed icons reappear and the expand control works without relaunching.

Written by the indexing model from the issue text.