Kelly–Postle local fractional Shearer conjecture

Let $G$ be a triangle-free graph, and let $d_G(v)$ denote the degree of a vertex $v\in V(G)$. Consider a probability distribution on the independent sets of $G$. Kelly–Postle's local fractional Shearer conjecture. There exists such a distribution in which every vertex $v\in V(G)$ appears with probability at least

$$(1-o(1))\frac{\ln d_G(v)}{d_G(v)}.$$

Here, the $o(1)$ term represents any function that tends to $0$ as the degree grows. The paper proves this conjecture, so the local form of Shearer's bound is solved.