Stability conjecture for independence polynomials of trees

Let $G$ be a tree, and let $i(G,x)$ denote its independence polynomial. A polynomial is stable here when all of its roots lie in the left half-plane. Tree stability conjecture. The independence polynomial $i(G,x)$ is stable; equivalently, all of its roots lie in the left half-plane. The source reports that the authors found no independence root of a tree in the right half-plane, but supplies no proof or resolution.