Local evacuation-shuffle algorithm for arbitrary skew tableaux

Let $T$ be any (semi)standard skew tableau and let $mathbox$ be an inner co-corner of $T$. Local evacuation-shuffle algorithm conjecture. There exists a local algorithm for computing $mathe sh(mathbox,T)$ without rectifying $T$ such that:

1. Each step exchanges $mathbox$ with an entry of $T$ of weakly increasing value.
2. The slide-equivalence class of $T$ is preserved throughout the algorithm.
3. The algorithm specializes to jeu de taquin when $T$ has straight shape and to $math{local\text{-}esh}$ when $T$ is ballot.

Each step should correspond, by conjugating with rectification, to a jeu de taquin slide of $mathbox$ through the rectification of $(mathbox,T)$. This proposes a uniform local description of evacuation-shuffle beyond the ballot and straight-shape cases; the existence of such an algorithm for arbitrary (semi)standard skew tableaux remains open.