Subcube extremality for the Hamming cube's beta-isoperimetric profile

Let $\mathcal{B}_\beta$ be the $\beta$-isoperimetric profile of the Hamming cube, defined for dyadic rationals $x\in\mathcal{Q}$ by

$$\mathcal{B}_\beta(x)=\inf_{n\ge 1}\inf_{\substack{A\subset\{0,1\}^n\\|A|=x}}\mathbf{E}h_A^\beta.$$

Here $h_A$ denotes the relevant Hamming-cube boundary function, and $k\ge 1$ is an integer.

Subcube extremality conjecture. For all $\beta\ge\frac12$ and all $k\ge 1$,

$$\mathcal{B}_\beta(2^{-k})=2^{-k}k^\beta.$$

The equality is attained by codimension-$k$ subcubes. Subcubes are known to be extremizers when $\beta=1$, while their extremality for $\beta<1$ is not otherwise known.