beta does not show which version is required and therefore marked as not compatible

@maxilein any ideas how this can happen?

Sorry, I don't. Then I would have made a suggestion.
I think the "recommended version" shown on your site's update page is determined by a combination of factors managed by the Drupal.org packaging system.

This is what gemini told me. And I think the hint may be in the first paragraph.

How the "Recommended Version" is Determined
Essentially, the recommended version is the latest stable release of a module on Drupal.org that is compatible with your site's major Drupal core version.

Here's what the module maintainer does, and how your site gets the information:

Code is Updated: The maintainer updates the module's code to be compatible with a new version of Drupal (like Drupal 11).

info.yml is Key: They update the core_version_requirement in the module's *.info.yml file to reflect this new compatibility. For a module to be compatible with Drupal 11, this line must include a constraint that matches, for example: core_version_requirement: '^10 || ^11'.

A New Release is Created: The maintainer then creates a new, tagged release on Drupal.org (e.g., 8.x-2.5, 2.0.0, etc.). When creating this release, they associate it with a specific Drupal core version.

Drupal.org's Update Service: Drupal.org's system reads the info.yml file from this new tagged release and notes its core compatibility. It then flags this new version as the latest stable, "recommended" release for that line of compatibility.

Your Site Checks In: When the Update Manager on your site contacts the Drupal.org update service, it reports your current Drupal core version. Drupal.org then responds by saying, "For your version of Drupal, here is the latest stable release of that module we have on record."

So, to directly answer your question:

There is nothing extra you need to add to the info.yml file to show a recommended version.

The only thing that matters in the info.yml file is the core_version_requirement key. Having a new, stable release on Drupal.org with an updated core_version_requirement is what makes it the "recommended" version for sites that are compatible with it. The Drupal.org infrastructure handles the rest.