In the conclusion of An Involution Principle-Free Proof of Stanley's Hook-Content Formula Krattenthaler notes that the techniques of the paper might be useful for finding bijective proofs of the enumeration of Symmetric Plane Partions. Has such a proof been found?

I am interested in generating random Symmetric Plane partitions with uniform probability faster than coupling from the past.

Any techniques for the other classes of Plane partitions would be of interest, as would similar techniques for shifted SSYT's. I know there is a version of Novelli-Pak-Stoyanovskii for shifted SYT's, something similar for SSYT's would be enlightening.