The generalized oscillating promotion conjecture for sl(n) web rotation

Assume there is a bijection between generalized oscillating tableaux of length $k$ with $n$ parts, whose final component is $(m,\ldots,m)$ for some $m\in\mathbb{Z}$, and $\operatorname{sl}(n)$ webs with $k$ boundary vertices. Let $D$ be an $\operatorname{sl}(n)$ web with a chosen leftmost vertex. Let $T(D)$ denote the generalized oscillating tableau associated with $D$, let $p(D)$ denote rotation of $D$ one vertex counterclockwise, and let $w(D)$ denote the word obtained from the states corresponding to the boundary vertices of $D$. Generalized oscillating promotion conjecture. For any such web $D$ with chosen leftmost vertex,

$$T(w(p(D))=p(T(w(D))).$$

That is, the generalized oscillating tableau associated with rotating $D$ is obtained by generalized oscillating promotion applied to the tableau associated with $D$. If true, this would identify generalized oscillating promotion with rotation of $\operatorname{sl}(n)$ webs, extending the known $\operatorname{sl}(3)$ correspondence to higher rank.