Galvin–McKinley–Perkins–Sarantis–Tetali zero-free region conjecture for linear hypergraphs

Let $G=(V,E)$ be a hypergraph, where a hypergraph is $k$-uniform if every edge has size $k$, and linear if each pair of edges intersects in at most one vertex. An independent set is a set of vertices containing no edge, and its independence polynomial is

$$Z_G(\lambda)=\sum_{I\in\cI(G)}\lambda^{\lvert I\rvert}.$$

The maximum degree of $G$ is denoted by $\Delta$.

Galvin–McKinley–Perkins–Sarantis–Tetali conjecture. For each $k\geq 2$, there exists a constant $C_k>0$ such that, if $G$ is a $k$-uniform linear hypergraph of maximum degree $\Delta$ and

$$\lvert\lambda\rvert\leq C_k\Delta^{-\frac{1}{k-1}},$$

then

$$Z_G(\lambda)\neq 0.$$

This conjecture proposes an improved zero-free region for the independence polynomial of linear hypergraphs. The paper's abstract states that it disproves this conjecture by constructing, for every $k\geq 3$, $k$-uniform linear hypergraphs with arbitrarily large maximum degree having a root of modulus $O\left(\frac{\log\Delta}{\Delta}\right)$, so the conjecture is refuted.