The cutoff conjectures for multi-stack and restricted random-to-random shuffles

Let $d$, $\mu$, $\gamma$, and $q$ be as in the paper's shuffle-chain theorem. Consider the multi-stack random-to-random shuffle chain $\mathbf{X}^{\operatorname{shuf}}$ and the restricted random-to-random shuffle chain $\mathbf{X}^{\operatorname{res-shuf}}$.

Multi-stack and restricted shuffle cutoff conjecture. Both chains exhibit cutoff, with the multi-stack chain around time

$$T\coloneqq n\log n\cdot\max\left\{\frac{1}{2\gamma},\frac{3}{4}\right\},$$

and the restricted chain around time

$$T\coloneqq n\log n\cdot\max\left\{\frac{1}{2\gamma},\frac{3}{4q}\right\}.$$

The conjecture is supported by the proof strategy in the paper and by the known $\frac{3}{4}n\log n$ cutoff for random-to-random shuffling, but the two stated generalisations remain open.