Polynomial chromatic-number conjecture for triangle-free intersection graphs of projective lines

An intersection graph of lines in $\mathbb{PR}^3$ has one vertex for each projective line and edges joining intersecting lines; it is triangle-free when it has no $3$-cycle. Polynomial chromatic-number conjecture for projective line graphs. There exists $\varepsilon>0$ such that, for every sufficiently large $n$, there exists a triangle-free intersection graph of $n$ lines in $\mathbb{PR}^3$ of chromatic number at least $n^{\varepsilon}$. Triangle-free projective-line intersection graphs of arbitrarily large chromatic number are known, but the asserted polynomial dependence on $n$ remains open; the source notes that affine constructions rely on parallel lines.