Poisson limit-profile conjecture for lazy conjugacy-class shuffles on the symmetric group

Let $P$ be the transition matrix of a lazy random walk on $S_n$ generated by a conjugacy class or a union of conjugacy classes having $n-o(n)$ fixed points. Suppose that $P$ exhibits total variation cutoff and $\ell^2$ cutoff at time $t_n$ with window $w_n$. Writing $\Phi_x(c)$ for the relevant total-variation profile, Poisson limit-profile conjecture.

$$\Phi_x(c)=d_{\operatorname{T.V.}}\bigl(\operatorname{Poiss}(1+e^{-c}),\operatorname{Poiss}(1)\bigr).$$

This conjecture predicts a universal Poisson total-variation profile for these conjugacy-class shuffles once both cutoff phenomena occur. The supplied text does not state whether the claim is known or open beyond presenting it in an open-questions section.