Asymptotic mixing-rate conjecture for large-alphabet polarizing kernels

For a prime-power alphabet size $q\geq2$ and a parameter $\beta\in(0,\frac{1}{2}]$, define

$$g_{0}(x)=(x(1-x))^{\beta}.$$

Let $\lambda_m$ denote the mixing-rate quantity used in the source, namely

$$\lambda_m=\sup_{x\in[0,1]}\frac{\overline{g}_{1}(x)}{g_{0}(x)}.$$

Asymptotic mixing-rate conjecture.

$$\lim_{m\to\infty}\frac{1}{\ln m}\ln\lambda_m =\lim_{m\to\infty}\frac{1}{\ln m}\ln\sup_{x\in[0,1]}\frac{\overline{g}_{1}(x)}{g_{0}(x)} =-\frac{1}{2}.$$

This is proposed as sufficient to obtain a sequence of inhomogeneous polar codes with near-optimal scaling for fixed $q$ as $m$ grows. The source supplies no resolution, so the conjecture remains open.