The design of the sub-browser is not to mirror the main browser completely, but to provide a secondary view of your collection. If you need to use the filter, consider reversing the two window functions; use the left (main) browser for your filter and use the right to find your playlists.
The reason the column settings are not inherited is users had requested the ability to customize the sub-browser column settings so they were always the same (but not matching those of the main browser).
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