The Robinson-Schensted correspondence is a bijection between elements of the symmetric group $S_n$ and pairs of standard tableaux of the same shape. The symmetric group is partially ordered by the Bruhat order, so this bijection induces a partial ordering "$\leq$"on the set of pairs of standard tableaux. Is there a natural "tableaux-theoretic" description of $\leq$? That is, can we give a necessary and sufficient condition for $(S,T)\leq (S',T')$ intrinsic to the set of pairs of tableaux?

(this was posted at mse a few weeks ago to no response)

Added later:
Each involution in $S_n$ is mapped to a pair $(P,P)$, so Bruhat order on involutions induces an order on standard tableaux. Characterising this order is listed as an `unsolved problem' in Björner and Brenti's *Combinatorics of Coxeter Groups.*