Berestycki's total-variation mixing conjecture for the giant component of the random transposition walk

Let $X_t$ be the random transposition walk on the symmetric group, let $G_t$ be the graph formed by the transpositions selected up to time $t$, and let $C_t$ be the largest connected component of $G_t$. Write $U_{\mathfrak{S}_{C_t}}$ for the uniform measure on the permutations of $C_t$, and let $d_{\operatorname{TV}}$ denote total-variation distance.

With notation as above, suppose $t \geq cn/2$ for $c > 1$. Berestycki's conjecture.

$$d_{\operatorname{TV}}(X_t|_{C_t}, U_{\mathfrak{S}_{C_t}}) = o_n(1).$$

The conjecture predicts that, once the associated random graph has entered the supercritical regime and its largest component is macroscopic, the restriction of the random transposition walk to that component is asymptotically uniform in total variation. Results on the cycle lengths of the largest cycles support this prediction, but the full total-variation statement remains open.