Hypergraph independence polynomials with a zero close to the origin

For each uniformity $k\ge 3$, we construct $k$-uniform linear hypergraphs $G$ with arbitrarily large maximum degree $\mathrm{\Delta }$ whose independence polynomial $Z_G$ has a root $\lambda$ with $\mid \lambda \mid =O\left(\frac{\log \mathrm{\Delta }}{\mathrm{\Delta }}\right)$. This disproves a recent conjecture of Galvin, McKinley, Perkins, Sarantis, and Tetali.