Webmentions are not handled properly by the comment blocks

Description

This was originally reported on core trac: https://core.trac.wordpress.org/ticket/59739
and the support forum https://wordpress.org/support/topic/separate-webmention-types-from-comments/

Because it concerns feature parity between the comments block and the classic comment type filters, I am moving it here.

With the new comments template for Twenty Twenty-Three, webmentions are not properly filtered: they appear both as comment and like or repost…
Expected result: if a comment is a like/repost webmention, the comment is hidden, but the like/repost webmention is shown accordingly.
I use the Webmention plugin by Pfefferle (+ IndieWeb and Syndication Links plugins). This means one of these plugins might probably have to handle all or part of the styling/behavior.

Step-by-step reproduction instructions

Reproduction steps were not provided by the reported, testing is needed.

Screenshots, screen recording, code snippet

No response

Environment info

No response

Please confirm that you have searched existing issues in the repo.

Yes

Please confirm that you have tested with all plugins deactivated except Gutenberg.

Yes

Start by reproducing the issue with the Twenty Twenty-Three comments template and the Webmention, IndieWeb, and Syndication Links plugins enabled, since no reproduction steps are provided. Compare the comments block behavior with the classic comment type filters; done means like/repost webmentions are hidden from comments and shown in their corresponding views.