Eligibility of an Android app to be "recognized" by Google Assistant

Question

I'm developing an Android app capable of voice interactions through the Assistant.

The app manifest exposes:

• an activity with android.intent.action.SEARCH action and android.intent.category.DEFAULT category, usually needed to allow opening the app by uttering "Hey, Google, open my-app";
• a shortcuts.xml file, referenced by a meta-data entity in the main activity manifest entry, in which I've attempted defining an actions.intent.OPEN_APP_FEATURE capability and a custom one.

After deploying the app via adb to my phone running Android 13, I can successfully invoke its capabilities via the App Actions Test Tool in Android Studio.

However, it seems that none of the "Hey Google" utterances that should result in opening it or invoking one of its capabilities works at all: Google Assistant keeps avoiding my app.

So, I wonder what are the conditions that allow an app to get "recognized" by Google Assistan.

In example: does it need to be published to Google Play? Does it also need to be a public app, or even a private one would suffice? What if it is published via, say, Samsung Store or third-party sources?

I didn't find any reference to this matter in the Android documentation. Anyway, I remember that at Android 7 ages it was really easy to open an app uttering "Hey Google, open my-app", even when not yet published...