Can't consistency override custom-search-result.html.twig theme template

With Custom Search (custom_search) Enabled 8.x-1.0-beta9 , suggestion disappear.
Do I need to repatch ?

Patch #15 doesn't apply to beta-9 and search-result.html.twig in my theme won't get used although it's the suggested file name.
So the patch needs at least a reroll.

I rerolled the patch from #15 and resolved merge conflicts in the rebase process which occurred in the custom_search_theme_registry_alter() function. Those conflicts were a result from replaced deprecated functions.
So I removed the lines just like patch #15 did.

The rerolled patch should apply to 8.x-1.0-beta9

Thanks to #19. It works for the beta9.
Search Custom Search (custom_search) Enabled 8.x-1.0-beta9

modules/contrib/custom_search$ curl https://www.drupal.org/files/issues/2023-04-24/custom_search-2885485-19.patch | git apply
  % Total    % Received % Xferd  Average Speed   Time    Time     Time  Current
                                 Dload  Upload   Total   Spent    Left  Speed
100 18063  100 18063    0     0  42601      0 --:--:-- --:--:-- --:--:-- 42601

I just tested the patch #19 on a fresh Drupal 10.1.5 with Custom Search 8.x-1.0-beta9 and I could override the template custom-search-result.html.twig by copying it from the module folder to my custom theme folder.
However, the twig debug is showing this error message in the markup :

<!-- THEME DEBUG -->
<!-- THEME HOOK: 'search_result' -->
<!-- FILE NAME SUGGESTIONS:
   * search-result--node-search.html.twig
   * search-result.html.twig
-->
<!-- INVALID FILE NAME SUGGESTIONS:
   See https://api.drupal.org/api/drupal/core!lib!Drupal!Core!Render!theme.api.php/function/hook_theme_suggestions_alter
   custom_search_result__node_search
   custom_search_result
-->
<!-- BEGIN OUTPUT from 'themes/thematik/templates/custom-search-result.html.twig' -->
[...]
<!-- END OUTPUT from 'themes/thematik/templates/custom-search-result.html.twig' -->

So I looked at the api doc mentionned and spoted this :

New suggestions must begin with the value of HOOK, followed by two underscores to be discoverable.

I tested to change the suggestions from "custom_search_result__" to "search_result__", it removes the warning and show the template used with an "x".
But I think that it's not the suggestion name that we want because it is the name of the core search result template.
Should we adopt a suggestion name like "search_result__custom_search__" ?
I'm wondering if these changes will break backward compatibility too.