Ensure panes update fake roots on updates

[philippjfr]

Fixes #6708

[codecov]

Codecov Report

Attention: Patch coverage is 75.00000% with 4 lines in your changes missing coverage. Please review.

Project coverage is 81.70%. Comparing base (65c0e24) to head (a5f4cbe).
Report is 1 commits behind head on main.

Files	Patch %	Lines
panel/pane/base.py	75.00%	4 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #6975      +/-   ##
==========================================
- Coverage   81.72%   81.70%   -0.03%     
==========================================
  Files         326      326              
  Lines       48055    48067      +12     
==========================================
- Hits        39274    39273       -1     
- Misses       8781     8794      +13     

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