OnMouse Events for Composite Collider with Rigidbody2d

I'm trying to assemble Tetriminos, made of smaller mino blocks, and add a collider to the parent object via a Composite Collider 2D.

For whatever reason, however, I cannot get any OnMouse Events to work on the parent object with a Rigidbody2D attached to it, for instance:

 using System.Collections;
 using System.Collections.Generic;
 using UnityEngine;
 
 public class DragRigid : MonoBehaviour {
 
      public void OnMouseEnter()
      {
          Debug.Log ("Started");
      }
 }
 

Does anyone see what the problem ist here? Are the setting for the Composite Collider 2D wrong, or the ones for the Rigidbody 2D? As far as I understood, OnMouse Events should also work in both cases, but in my case they don't. This seems to be so simple, yet I have searched everywhere and I can't find the problem.

I'd appreciate any input!