Cutoff scale for biased one-sided transposition shuffles with general weights

Let $\operatorname{OST}_{n,w}$ be the biased one-sided transposition shuffle, where $w(j)/j$ is monotonically decreasing, and let $N_w(n)$ and $N'_w(n)$ be the quantities used in the source. Cutoff conjecture. The shuffle $\operatorname{OST}_{n,w}$ exhibits a total variation cutoff at time

$$\left(\frac{N_w(n)}{N'_w(n)}\right)n\log n.$$

A lower bound of this scale is given, while the source states that proving the corresponding upper bound remains unresolved.