Density conjecture for chromatic roots of the graphs $S_{m,n}$

Let $S_{m,n}$ be the graph family considered in the paper, and let a chromatic root mean a zero of its chromatic polynomial in the complex $q$-plane. There exists a finite constant $Q$ such that

$$|q|>Q$$

is filled densely by chromatic roots of $S_{m,n}$ as $m,n\to\infty$. Chromatic-root density conjecture. There exists a constant $Q<\infty$ such that the chromatic roots of the graphs $S_{m,n}$ become dense in the region $|q|>Q$ when $m,n\to\infty$. The conjecture concerns uniformity in $m$; the paper emphasizes that the available fixed-$m$ asymptotic results do not imply it because their error bounds need not be uniform. Thus the claim remains open.