>     > Given that text properties don't nest, I matched the faces and use
>     separate
>     > mouse-face properties for each component so they could be highlighted
>     > separately.  Look at the attached screenshots.  You still might not
>     like
>     > it.  I also handled the case where the close button might be text
>     only vs.
>     > icon so it works for both.
>
>     Does using 'add-face-text-property' (like in
>     'tab-bar-tab-name-format-face')
>     instead of 'propertize' help to make the close button look like part of
>     the tab?
>
> Nope.  I think the issue is that the mouse-face on the caption left of the
> close button cannot itself highlight the whole string while simultaneously
> having a mouse-face on the close button.  So we either live with the visual
> separation which appears only when people mouse over caption segments, or
> abandon mouse-face on the close button.  I prefer the visual separation as
> it makes it clear the close button is separate from the caption text.

Unfortunately, the visual separation is too ugly.  But maybe
mouse-face on the close button is not needed, because hovering
over the close button already has visual indication in the form
of the raised button.