Concavity conjecture for polar-code interpolation functions

For a prime-power alphabet size $q\geq2$, an integer kernel dimension $m\geq2$, and a parameter $\beta\in(0,1)$, define

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

Concavity conjecture. The function $\overline{g}_{n}(x)$ is concave on $[0,1]$ for every $n$.

If true, this would make the scaling analysis of the associated sequence of inhomogeneous polar codes rigorous in terms of $\lambda_m$. The source presents this as unproved; the notation $\overline{g}_n$ is not defined in the supplied span.