Gumbel separation-profile conjecture for the random transposition shuffle

Consider, for $n\geq 2$, the random transposition shuffle $(X_t^{(n)})_{t\geq 0}$ on the symmetric group $\mathfrak{S}_n$, in discrete or continuous time, and let $\mathrm{d}_{\textup{sep}}^{(n)}(t)$ denote its separation distance to stationarity at time $t\geq 0$. For any $c\in\mathbb{R}$, the Gumbel separation-profile conjecture asserts that

$$\mathrm{d}_{\textup{sep}}^{(n)}\left(\left\lfloor \frac{1}{2}n(\ln n+c)\right\rfloor\right)\xrightarrow[n\to\infty]{}1-e^{-e^{-c}}.$$

In words, the separation profile for random transpositions should be given by a Gumbel distribution. The conjecture is motivated by the likelihood calculation for $n$-cycles and would follow from a positive answer to the question of whether, for every time, the minimum of likelihood is attained at $n$-cycles. Its resolution is not supplied here.