The "yes" answer follows immediately from [Theorem 2.5 in this paper by Kusuoka][1].


  [1]: https://reader.elsevier.com/reader/sd/pii/S0304414916300850?token=83AF411073D65047F42CC4F378ABD1D841ADA751E9CEE4DCE66CD933F7D9311AC9D96047079D3456564E461FAA7C60B7&originRegion=us-east-1&originCreation=20211007170037