Biased-measure entropy/influence conjecture

Let $\mu_p$ be the product measure on the discrete cube $\{0,1\}^n$, and let $f:\{0,1\}^n_p\to\{-1,1\}$ be a Boolean function. Write $\operatorname{Ent}_p(f)$ for its spectral entropy and $\operatorname{I}_p(f)$ for its total influence. Biased-measure entropy/influence conjecture. There exists a universal constant $c$ such that, for every $0<p<1$, every $n$, and every such $f$,

$$\operatorname{Ent}_p(f)\leq c p\log(1/p)\,\operatorname{I}_p(f).$$

The paper proves that this statement follows from the original uniform-measure conjecture and says the bound is tight for the graph property of containing a fixed-size clique at the critical probability. It is presented as an open conjectural generalization.