Polynomial row-treewidth conjecture for string graphs on fixed surfaces

Let $G$ be a string graph drawn in a surface of fixed Euler genus $g$, and let $\Delta$ be the maximum degree of $G$. Polynomial row-treewidth conjecture. The row treewidth of $G$ is at most $f(\Delta,g)$ for some polynomial function $f$. The conjecture seeks to improve the paper's exponential or larger bounds for string graphs of bounded maximum degree; the source treats it as an open problem.