Binary-entropy lower-bound conjecture for Fourier entropy on the biased cube

Let $f:(\{0,1\}^n,\mu_p^n)\to\{\pm1\}$ be Boolean-valued and let $0<p<1$. Define

$$q:=4p(1-p)\in[0,1]$$

and let $h(q)=-q\log q-(1-q)\log(1-q)$ be the binary entropy function. Binary-entropy lower-bound conjecture. For every such $f$ and $p$,

$${\rm Ent}_p(f) \ge h(q)\cdot \sum_{k=1}^n {\rm Inf}_k^{(p)}[f]^2.$$

The conjecture proposes the optimal bias-dependent constant in the paper's lower bound; the supplied text gives no resolution.