Order-q conjecture for K-promotion on three-row rectangular increasing tableaux

An increasing tableau of shape $\lambda$ with entries in $[q]\coloneqq\{1,\ldots,q\}$ is a filling of the Young diagram of $\lambda$ with elements of $[q]$ whose rows and columns are strictly increasing; write $\operatorname{Inc}^q(\lambda)$ for the set of such tableaux. Let $\operatorname{\mathcal{P}}$ denote $K$-promotion. Order-$q$ conjecture. For every $T\in\operatorname{Inc}^q(3\times c)$, where $T$ is a three-row rectangular increasing tableau, one has

$$\operatorname{\mathcal{P}}^q(T)=T.$$

The order of $K$-promotion divides $q$ for rectangular shapes with at most two rows, whereas the three-row case remains open and is only partially understood.