Toggling the rule `@@||example.org^$important,document` doesn't properly update the Browser Assistant icon

#### :wave: Hello,

Please find below two different but related problems.
Ensure a clean state between the two.

> __Note__: Issue opened in **CoreLibs** instead of **Browser Assistant**,
based on my previous ones [here](https://github.com/AdguardTeam/CoreLibs/issues/1545) and [there](https://github.com/AdguardTeam/CoreLibs/issues/1546) (which had been moved to **CoreLibs**).

Thanks in advance! :v:
 
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##### :computer: Environment
**AdGuard Browser Assistant:** Beta 1.3.32 (on Firefox 111.0.1)
**AdGuard:** 7.13 nightly 5 (build 4224, CL 1.11.69)
**Windows:** 10 Pro 22H2 (OS Build 19045.2728)