And even another question: in MvcFunctionBase there are two methods ExecuteAsync and ExecuteRouteAsync. Additionally, the MvcControllerFunction appears to do some check on the methods of a controller class if there are any async methods.
Anyway: whenever I try to use the async method of my controller an exception is thrown

System.InvalidOperationException: The asynchronous action method 'Index' returns a Task, which cannot be executed synchronously.

The stack trace is indicating, that instead of ExecuteAsync the synchronous method "Execute" was called.

Are async controller actions yet not supported?