Rapid mixing for standard set-valued tableaux with bounded diagonal complexity

Let $\mathcal{SVT}$ denote the set of standard set-valued tableaux, and let $\mathcal{MC}_{\sf SVT}(\lambda,N,S)$ be the associated Markov chain. For a tableau shape $\lambda$, prescribed entries $N$, and subset $S$, write $\lambda\setminus S$ for the boxes of $\lambda$ outside $S$, let $(\lambda\setminus S)^+$ denote the corresponding augmented shape, and let $\sf sv$ be the statistic used to measure its relevant row-and-column complexity. For $d\in\mathbb{N}$, define

$$\mathcal{F}_d=\{(\lambda,N,S)\in\mathcal{SVT}:{\sf sv}((\lambda\setminus S)^+)\leq d\}.$$

Bounded-complexity rapid-mixing conjecture. The Markov chain $\mathcal{MC}_{\sf SVT}(\lambda,N,S)$ is rapidly mixing for every $(\lambda,N,S)\in\mathcal{F}_d$.

The conjecture proposes extending the paper's rapid-mixing result beyond instances with at most two relevant long rows or columns. For each fixed $d$, it predicts rapid mixing throughout the class whose statistic ${\sf sv}((\lambda\setminus S)^+)$ is bounded by $d$; the supplied text gives no resolution of the conjecture.