The model-free string-graph coloring algorithm conjecture

Let $G$ be a string graph, let $n$ be its size, and let $\omega(G)$ and $\beta(G)$ denote the parameters used by the source. An $f(\omega(G),\beta(G))$-coloring is a coloring with the stated number of colors. Model-free string-graph coloring conjecture. There are functions $f,C\colon\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ and an algorithm that takes a graph $G$ of size $n$ as input and, in $n^{C(\omega(G),\beta(G))}$ time, either computes an $f(\omega(G),\beta(G))$-coloring of $G$ or correctly reports that $G$ is not a string graph. The source previously describes this algorithm when an intersection model is supplied; removing that requirement is left as an open conjecture.