Linear vertex deletion conjecture for bounded boxicity of surface graphs

Let $G$ be a graph embedded on a surface of Euler genus $g$. The boxicity of a graph is the minimum dimension in which it can be represented as the intersection graph of axis-parallel boxes.

Linear vertex deletion conjecture. There is a constant $c>0$ such that, for every graph $G$ embedded on a surface of Euler genus $g$, at most $cg$ vertices can be removed so that the resulting graph has boxicity at most $3$.

The conjecture is a surface-graph variant of the preceding conjecture for locally planar graphs. The supplied source gives no evidence that it has been resolved; it would in particular provide a linear-size deletion bound depending on the Euler genus.