Rowmotion-compatible direct bijection

Construct a direct bijection proving the equality of descent sets between the row- and column-labeled linear extensions of an arbitrary finite poset, or at least between ascent sets and high-descent sets of standard Young tableaux of any partition shape, such that its restriction to \(\SYT(n^2)\) coincides with the rowmotion map \(\usualrowmotion\) induced from order ideals of the type \(A\) root poset.

Background

The paper presents two candidates for a rowmotion operation on standard Young tableaux. One is the explicitly defined map on rectangular tableaux that agrees with the usual rowmotion map on $\SYT(n^2)$ and preserves the equality of the number of ascents and high descents, but not generally the full ascent and high-descent sets when $k\ge3$.

The second candidate is a recursively defined bijection that matches the full sets $\Asc(T)$ and $\HDes(f(T))$ for arbitrary shapes, but it does not agree with the usual rowmotion map on $\SYT(n^2)$. The open problem asks for a direct construction resolving this incompatibility.