Thomassen's Hamiltonian graph conjecture

Let $G$ be a hamiltonian loopless graph with $n$ vertices, and let $P(G,t)$ denote its chromatic polynomial. Thomassen's conjecture.

$$P(G,t)\text{ is non-zero with sign }(-1)^n\quad\text{for }t\in(1,2).$$

The conjecture extends the known zero-free interval for graphs with a Hamilton path, where a smaller interval ending near $1.296$ is available. A smallest counterexample would be 3-connected, and the conjecture would follow from the preceding 3-connected-graph conjecture.