Transpose bijection conjecture for catabolizable tableaux

Let $R$ be a sequence of rectangles, and let $CT(\lambda;R)$ and $CCT(\lambda^t;R^t)$ denote the row- and column-catabolizable tableaux defined in the source. Catabolizable transpose conjecture. When both $R$ and $R^t$ are dominant, the proposed map gives a bijection

$$CT(\lambda;R)\longrightarrow CCT(\lambda^t;R^t).$$

This would relate the two forms of catabolizability under transposition; the source identifies column strictness of the constructed tableau as the essential point still requiring verification.