Higuchi's conjecture on positive combinatorial curvature

Let a tessellation be a graph embedded in the plane so that every face is a polygon and the embedding satisfies the usual tessellation conditions. At each vertex $v$, let its combinatorial curvature be

$$\kappa(v)=1-\frac{d_v}{2}+\sum_{f\sim v}\frac{1}{d_f},$$

where $d_v$ is the degree of $v$ and $d_f$ is the degree of the face $f$. Higuchi's conjecture. If $\kappa(v)>0$ for every vertex $v$, then the tessellation is a finite graph. This is proposed as a discrete analogue of a weak form of the Bonnet–Myers theorem; the source gives no resolution of the conjecture.