The asymptotic mixing-time conjecture for the $S_k$ shuffle

Let $N$ be the number of positions and let $k=o(N^{1/2})$. Denote by $t_{\mathrm{mix}}(\varepsilon)$ and $t_{\mathrm{mix}}^{\prime}(\varepsilon)$ the $4\varepsilon$4-mixing times of the $S_k$ shuffle with and without boundary, respectively. Mixing-time conjecture. For every $4\varepsilon\in(0,1)4$,

$$\lim_{N\rightarrow\infty}\frac{k(k^2-1)t_{\mathrm{mix}}(\varepsilon)}{N^2}=\lim_{N\rightarrow\infty}\frac{k(k^2-1)t_{\mathrm{mix}}^{\prime}(\varepsilon)}{N^2}=\frac{6}{\pi^2}.$$

Equivalently, both versions of the $S_k$ shuffle are conjectured to exhibit cutoff in this regime. The statement concerns the asymptotic mixing-time constant when $k=o(N^{1/2})$; the supplied text does not indicate that it has been proved or disproved.