Salas–Sokal conjecture for planar bipartite graphs

Let $G$ be a planar bipartite graph, and let $P(G,t)$ be its chromatic polynomial. Let $\tau=(1+\sqrt{5})/2$ be the golden ratio. Salas–Sokal conjecture.

$$P(G,t)>0\quad\text{for }t\in[\tau^2,\infty).$$

Equivalently, planar bipartite graphs should have no real chromatic roots at or above $\tau^2$. The source notes that Salas and Sokal constructed families with chromatic roots tending to $\tau^2$, giving evidence for sharpness.