Friedgut's continuous-cube resilience conjecture

Let $f\colon [0,1]^n \to \{0,1\}$ be a measurable function, and let $I_S^b(f)$ denote the corresponding resilience quantity for a set $S\subseteq[n]$ and $b\in\{0,1\}$. Friedgut's conjecture. There exists a set $S$ of size $o_\epsilon(n)$ such that

$$I_S^b(f) \ge 1-\epsilon$$

for some $b\in\{0,1\}$. This conjecture seeks a correct continuous-cube analogue of the false consequence previously attributed to the generalized KKL inequality; the source gives no resolution.