Separation cutoff for biased one-sided transposition shuffles

Let $\operatorname{OST}_{n,\alpha}$ be the biased one-sided transposition shuffle with weight parameter $\alpha\in(0,\infty)$, and let $t_{n,\alpha}$ be the total-variation cutoff scale defined in the source. Separation-cutoff conjecture. The shuffle exhibits a cutoff in separation distance at time

$$t_{n,\alpha}\log n.$$

The source explains that this is conjectured because the available strong stationary-time argument does not give a matching upper bound; the claimed time is the total-variation cutoff scale.