Floccus not usable in Private Window (Firefox)

### Which version of floccus are you using?

4.17.1

### Sync method

Nextcloud Bookmarks

### Which browser are you using? In case you are using the phone App, specify the Android or iOS version and device please.

Firefox v107.0

### Which version of Nextcloud Bookmarks are you using? (if relevant)

v11.0.4

### Which version of Nextcloud? (if relevant)

v25.0.1

### What kind of WebDAV server are you using? (if relevant)

_No response_

### Describe the Bug

If you try to use Floccus in a private window of Firefox, Floccus isn't working. For example click on the Extension icon in your toolbar and the floccus window opens, but only shows a spinner. You have to open a non-private-window (normal window) to use Floccus and to be able to sync.

Floccus Extension is allowed to run in the private mode of firefox in the Firefox Extension Settings (Plugins list) already.

See also the comment from last year of this closed issue: https://github.com/floccusaddon/floccus/issues/622#issuecomment-878862701

### Expected Behavior

Same behaviour as in a general firefox window

### To Reproduce

Open a private window in Firefox and try to use Floccus.

### Debug log provided

- [ ] I have provided a debug log file

Start by reproducing the spinner in Firefox v107.0 private browsing with Floccus 4.17.1 and compare it with a normal window, using the extension icon and sync flow described here. Trace the private-window startup and sync behavior; done means the extension opens and syncs in private windows as it does normally.

Written by the indexing model from the issue text.