BlendShapes with animations from Blender

I'm trying to understand how animations with BlendShapes work in Unity and Blender. I'm using Unity 5.1.1f1 and Blender 2.74.

I have a simple dancing cube with 1 animation and 2 BlendShapes. The BlendShapes are working well until I turn on the animator component. Then only the animation is played and the BlendShapes are completely ignored.

I have the following code controlling the cube.

 using UnityEngine;
 
 public class CubeController : MonoBehaviour {
 
     Animator animator;
 
     private SkinnedMeshRenderer skinnedMeshRenderer;
     private Mesh skinnedMesh;
     int blendShapeCount = 0;
 
     private MeshCollider meshCollider;
 
     [SerializeField] GameObject cube;
     [SerializeField][Range(0, 1)] float size;
     [SerializeField][Range(0, 1)] float deformation;
 
     void Awake()
     {
         animator = GetComponent<Animator>();
 
         // Using BlendShapes
         skinnedMeshRenderer = cube.GetComponent<SkinnedMeshRenderer>();
         skinnedMesh = skinnedMeshRenderer.sharedMesh;
 
         meshCollider = GetComponent<MeshCollider>();
     }
 
 
     // Use this for initialization
     void Start () {
 
         blendShapeCount = skinnedMesh.blendShapeCount;
         Debug.Log("blendshape count: " + blendShapeCount);
         if (blendShapeCount == 2)
         {
             skinnedMeshRenderer.SetBlendShapeWeight(0, size * 100);
             skinnedMeshRenderer.SetBlendShapeWeight(1, deformation * 100);
         }
 
         meshCollider.sharedMesh = skinnedMesh;
 
     }
 
 }

Here you have the exported dancing cube .fbx: https://drive.google.com/file/d/0B3ELWKyJ3vaOU0oxeXhWTFBXdEU/view?usp=sharing

Are BlendShapes with animations supported?

Thanks!