Cutoff for biased one-sided transposition shuffles on the hyperoctahedral group

Let $\operatorname{OST}_{n,w}$ be the biased one-sided transposition shuffle, where $w(j)/j$ is monotonically decreasing, and let $N_w(n)$ denote its normalising quantity. Cutoff conjecture. The shuffle $\operatorname{OST}_{n,w}$ exhibits a total variation cutoff at time

$$\left(\frac{N_w(n)}{w(n)}\right)\log n.$$

The preceding upper bound is proved, but the matching lower bound is not established for this class of weight functions.