Vertex-minimal maximum-excess triangulation conjecture

Let $\Sigma$ be a surface, let $\omega$ be the largest integer such that $K_\omega$ embeds in $\Sigma$, and call a triangulation vertex-minimal if it has the minimum number of vertices among triangulations of $\Sigma$. An irreducible triangulation is a triangulation with no contractible edge, and the excess is the quantity defined in the paper.

Vertex-minimality conjecture. For every surface $\Sigma$, the maximum excess is attained by some vertex-minimal triangulation of $\Sigma$ that contains $K_\omega$ as a subgraph. Moreover, if

$$\Sigma\notin\{\mathbb{N}_2,\mathbb{N}_3\},$$

then every irreducible triangulation with maximum excess is vertex-minimal and contains $K_\omega$ as a subgraph.

The source gives the vertex orders of vertex-minimal triangulations and presents this as a final strengthening of the preceding complete-subgraph conjecture. Its general validity remains open.