Linear Euler-characteristic bound for nonnegative-curvature vertices in polyhedral maps

Let $G$ be a simple polyhedral map on a surface $\mathbb{M}$ with Euler characteristic $\chi(\mathbb{M})\le 0$. For each vertex $v$, let $F(v)$ be the set of incident faces and define its combinatorial curvature by

$$\Phi(v)=1-\frac{\operatorname{deg}(v)}{2}+\sum_{\alpha\in F(v)}\frac{1}{\operatorname{deg}(\alpha)}.$$

The conjecture. For each surface $\mathbb{M}$, there exists a constant $c_{\mathbb{M}}$ such that, if

$$|V(G)|>c_{\mathbb{M}}|\chi(\mathbb{M})|,$$

then $G$ contains a vertex $v$ with $\Phi(v)\ge 0$.

This is presented as an expected consequence for polyhedral maps on a fixed surface. The supplied text gives no resolution or further supporting result, so its status remains open.