Polynomial chromatic-number conjecture for high-girth intersection graphs of lines

Let $g\in\mathbb{N}$. An intersection graph of lines in $\mathbb{R}^3$ has one vertex for each line and edges joining intersecting lines; its girth is the length of its shortest cycle. Polynomial chromatic-number conjecture for high-girth line graphs. For every $g\in\mathbb{N}$ there exists $\varepsilon>0$ such that, for every sufficiently large $n$, there exists an intersection graph of $n$ lines in $\mathbb{R}^3$ of girth at least $g$ and chromatic number at least $n^{\varepsilon}$. Davies's result gives arbitrarily large chromatic number at fixed girth, while this conjecture asks for polynomial growth in the number of lines.