Sokal's half-plane conjecture for chromatic roots

Let $G$ be a graph, let $\Delta(G)$ denote its maximum degree, and let $P(G,t)$ be its chromatic polynomial. Sokal's conjecture.

$$|P(G,t)|>0$$

for every complex number $t$ satisfying

$$\operatorname{Re}(t)>\Delta(G).$$

This would strengthen the known zero-free exterior disc bound $|t|\geq C\Delta(G)$ and would be a half-plane analogue of Brooks's theorem. The source presents it as a possible extension and gives no resolution.