Localised representation conjecture for string graphs on surfaces

Let $\Sigma$ be a surface with Euler genus $g$, and let $G$ be a string graph in $\Sigma$. A string representation $\mathcal{C}$ of $G$ in $\Sigma$ assigns a non-self-intersecting curve to each vertex, with two curves intersecting exactly when the corresponding vertices are adjacent. Localised representation conjecture. There exists a function $f$ such that for every surface $\Sigma$ with Euler genus $g$ and every string graph $G$ in $\Sigma$, there is a string representation $\mathcal{C}$ of $G$ in $\Sigma$ such that, for each $u\in V(G)$, the curve representing $u$ is involved in at most $f(d_G(u),g)$ crossings. The planar case is established earlier in the paper, while the extension to arbitrary surfaces is presented as open and would yield bounded-degree consequences for string graphs on surfaces.