Fourier concentration conjecture for low-influence Boolean functions

Fix $0<p<1$. Let $f:\{-1,1\}^n_p\to\{-1,1\}$ be a Boolean function, let $\operatorname{I}(f)$ denote its total influence, and let $\hat f(S)$ be its Fourier coefficients. Fourier concentration conjecture. There exists a constant $C(p)>0$ such that, for every $\epsilon>0$, every $n$, and every such $f$, there is a set $B_\epsilon\subset\{0,1\}^n$ satisfying

$$\log|B_\epsilon|\leq C(p)\,\operatorname{I}(f)$$

and

$$\sum_{S\notin B_\epsilon}\hat f(S)^2<\epsilon.$$

This conjecture would strengthen the Fourier-concentration conclusion in Friedgut's low-influence characterization: most Fourier weight would lie on at most exponentially many coefficients in the total influence. The source presents it as an open conjecture resembling Bourgain–Kalai's theorem.