A local evacuation-shuffling algorithm for arbitrary skew tableaux

Let $T$ be any (semi)standard skew tableau and let $\scalebox{.5}{\young(x)}$ be an inner co-corner of $T$. Write $\operatorname{esh}$ for evacuation-shuffling, $\operatorname{rect}$ for rectification, and $\operatorname{local\text{-}esh}$ for the local evacuation-shuffling algorithm on ballot tableaux. Local evacuation-shuffling conjecture. There exists a local algorithm for computing $\operatorname{esh}(\scalebox{.5}{\young(x)},T)$ without rectifying $T$ such that each step exchanges $\scalebox{.5}{\young(x)}$ with an entry of $T$ of weakly increasing value, preserves the Knuth equivalence class of the word of $T$ with $\scalebox{.5}{\young(x)}$ omitted, and specializes to jeu de taquin when $T$ has straight shape and to $\operatorname{local\text{-}esh}$ when $T$ is ballot. Each step should correspond, under conjugation by rectification, to a jeu de taquin slide of $\scalebox{.5}{\young(x)}$ through $\operatorname{rect}(\scalebox{.5}{\young(x)},T)$.