Talagrand's bounded-uniformity fractional expectation threshold conjecture

Let $X$ be a finite set, let $k\in\mathbb{N}$ satisfy $k\leq\log_L|X|$, and let $g\colon{\mleft(\kern-.1em\genfrac{}{}{0pt}{}{X}{k}\kern-.1em\mright)}\to[0,1]$. Define $\left<g\right>$ by

\left<g\right>:=\left\{S\in 2^X\mathrel{\left|}\sum_{T\subseteq S}g(T)\geq 1\right.\right\},

and for a family $G\subseteq 2^X$ define $w(G,p):=\sum_{S\in G}p^{|S|}$, with $w(g,p)$ defined analogously. The bounded-uniformity conjecture. There exists a fixed $L>1$ such that, for all finite sets $X$, all such $k$, all such functions $g$, and all $p\in[0,1]$, if $w(g,p)=1$, then there exists a set $G\subseteq2^X\setminus\{\emptyset\}$ such that $\left<g\right>\subseteq\left<G\right>$ and

$$w\left(G,\frac{p}{L}\right)\leq1.$$

This is a restricted form of Talagrand's conjecture whose resolution would imply the full conjecture. Results cited in the source cover constant-size support, and current quantitative bounds do not establish this claim in general.