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on Tableaux problem 3

The statement

3.[Wor] Consider the tree $SYT$ of (unshifted) standard Young tableaux, with the tree structure generated by containment. For any node with a symmetric shape, there is an automorphism of $SYT$ (as an unlabeled tree) that transposes all nodes in the subtree rooted at the node and leaves all other nodes fixed. Computation suggests that all automorphisms of $SYT$ are compositions of these transpose automorphisms (with the identity being the composition of zero transpose automorphisms). Computation further suggests that each node of $SYT$ is determined up to transpose by the census (cardinalities) of the ranks of the subtree rooted at the node.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.