Beraha's conjecture on chromatic roots near the Beraha numbers

For each integer $r\geq2$, define the Beraha number

$$b_r=2+2\cos\frac{2\pi}{r}.$$

A plane triangulation is a loopless plane graph whose faces all have size three. Beraha's conjecture. For every $r\geq2$ and every $\epsilon>0$, there exists a plane triangulation with a real chromatic root in

$$(b_r-\epsilon,b_r+\epsilon).$$

This would imply the existence of plane triangulations with real chromatic roots arbitrarily close to $4$, since $b_r$ tends to $4$. The source says that this remains open, while complex roots arbitrarily close to $4$ are known.