The $(2,0)$-tight spanning subgraph characterization of rigidity in $S_g$

Let $g\geq 2$, let $S_g$ be the closed orientable surface of genus $g$, and let a simple graph be called rigid in $S_g$ when it has a well-positioned realisation whose associated geodesic framework is both regular and continuously rigid. A graph is rigid in $S_g$ if and only if it contains a $(2,0)$-tight spanning subgraph.

This conjecture proposes a combinatorial characterization of the graph family arising from shortest-geodesic rigidity on surfaces. The preceding discussion notes that this notion is not generic, because different well-positioned realisations can yield different gain maps; the conjectured characterization therefore concerns the existence of a suitable realisation.