Spanning triangulated subsurface conjecture for triangulations of surfaces

Let $\mathbb{S}_g$ denote the surface with Euler genus $g$ and two boundary components, with the cases $\mathbb{S}_1$ and $\mathbb{S}_2$ being a disk and a cylinder, respectively. A triangulation of a surface is an embedded graph whose boundary lies in the graph and whose faces are open disks bounded by copies of $K_3$. A subgraph is spanning if it contains every vertex of the original graph.

Spanning triangulated subsurface conjecture. For all $g \geq 0$, every triangulation of a surface of Euler genus $g$ contains a spanning subgraph which is a triangulated $\mathbb{S}_g$.

This conjecture generalizes the known results that triangulations of the projective plane contain spanning triangulated disks and triangulations of the torus contain spanning triangulated cylinders. It is also described as a strengthening of a conjecture of Nevo and Tarabykin that every triangulation of a surface contains a spanning subgraph that is planar and Laman; its general status is not specified in the source.