Thomassen's conjecture on real chromatic roots of planar graphs

Let $G$ be a loopless planar graph, and let $P(G,t)$ be its chromatic polynomial. Thomassen's conjecture. The real chromatic roots of planar graphs are dense everywhere in

$$\left[\frac{32}{27},4\right].$$

Real chromatic roots are already known to be dense in $[32/27,3]$. Extending density to $4$ would show that planar graphs have real chromatic roots arbitrarily close to the four-colour threshold.