Periodicity conjecture for increasing tableaux of shape $3 \times n$

Let $\operatorname{Inc}^q(3 \times n)$ denote the increasing tableaux of shape $3 \times n$ with entries from an alphabet of size $q$, and let $\mathcal{P}$ denote $K$-promotion on these tableaux.

Periodicity conjecture. For every $T \in \operatorname{Inc}^q(3 \times n)$,

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

This conjecture was posed in earlier work with K. Dilks and J. Striker. The paper proves the analogous statement for tableaux of rectangular shape after restricting to the frame, and notes that the full assertion is already known for shape $2 \times n$; the $3 \times n$ case is the next unresolved case described here.