Bishop volume comparison conjecture for locally tessellating planar graphs

Let $p,q\ge 3$ with $1/p+1/q\le 1/2$ be given. Let ${\cal G}=({\cal V},{\cal E},{\cal F})$ be a locally tessellating planar graph without cut locus, meaning that it has no cut locus, and suppose that

$$|v|\le p,\qquad |f|\le q$$

for all vertices $v\in{\cal V}$ and faces $f\in{\cal F}$. Bishop volume comparison conjecture. Then

$$\mu({\cal G})\le \mu({\cal G}_{p,q}).$$

Here ${\cal G}_{p,q}$ is the regular plane tessellation with vertex degree $p$ and face degree $q$, and $\mu$ denotes the exponential growth rate. This is proposed as a discrete analogue of the Bishop volume comparison theorem for Riemannian manifolds; the preceding theorem proves related bounds under a vertex-degree bound alone or together with face-regularity, while the general simultaneous vertex- and face-degree statement remains open.