Limit-profile conjecture for the strictly biased random transposition shuffle

Let $P$ be the transition matrix of the biased random transposition shuffle on $N$ cards, let $U$ be the uniform distribution on permutations, let $d_{\operatorname{TV}}$ denote total variation distance, and let $\operatorname{Poiss}(\lambda)$ denote the Poisson distribution with mean $\lambda$. For $c \in \mathbb{R}$ and $b<1$, consider the shuffle after $\frac{1}{2b}N(\log N-c)$ steps. Limit-profile conjecture. The total variation distance satisfies

$$d_{\operatorname{TV}}\left(P^{\frac{1}{2b}N(\log N-c)},U\right)\to d_{\operatorname{TV}}\left(\operatorname{Poiss}\left(1+\frac{e^c}{2}\right),\operatorname{Poiss}(1)\right),\qquad\text{as }N\to\infty.$$

This conjectures a cutoff limit profile for the strictly biased random transposition shuffle, based on the limiting distribution of the number of fixed cards. It contrasts with the unbiased random-transposition case, where the corresponding limiting Poisson mean is $1+e^c$; the claimed profile remains unproved in the source.