Keller–Mossel–Schlank Fourier-Entropy-Influence conjecture for the biased cube

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

$${\rm Ent}_p(f)=\sum_{S\subset[n]}\hat f(S)^2\log\left(\frac{1}{\hat f(S)^2}\right)$$

and ${\rm I}^{(p)}[f]=\sum_{i=1}^n{\rm Inf}_i^{(p)}[f]$. Keller–Mossel–Schlank's biased FEI conjecture. There exists a universal constant $C$ such that

$${\rm Ent}_p(f)\le Cp\log\left(\frac{1}{p}\right)\cdot{\rm I}^{(p)}[f].$$

This is the $p$-biased generalization of the Fourier-Entropy-Influence conjecture; its resolution is not established by the supplied source.