Cutoff conjecture for the Burnside process

Fix $k\ge 2$, and for each $n$ let $K_n$ be the Burnside process on $(C_k^n,S_n)$. Consider starting states whose limiting empirical distribution on $C_k$ assigns a positive proportion to at least two different values. Burnside-process cutoff conjecture. The chain $K_n$ has cutoff in both $\ell^1$ and $\ell^2$ when started from such states. The paper proves that $n/\log n$ steps are necessary for $\ell^2$ mixing from many such starting states, but lacks the spectral information needed for matching upper bounds; the conjecture asserts the resulting cutoff in both norms.