Uniform dimension threshold conjecture for positive-genus tilings

Let $g>0$. A simple tiling $\mathcal T$ of a genus-$g$ surface has an associated incidence theorem over a division ring ${\mathrm D}$, and $n$ denotes the dimension of the ambient projective space. Uniform dimension threshold conjecture. There exists a constant $n_g$ such that, for any simple tiling $\mathcal T$ of a genus-$g$ surface, the corresponding incidence theorem over ${\mathrm D}$ is equivalent to commutativity of ${\mathrm D}$ in every dimension $n\geq n_g$. In particular, $n_1=3$, so any theorem in dimension $n\geq3$ corresponding to a tiling of the torus holds over ${\mathrm D}$ if and only if ${\mathrm D}$ is commutative. This is presented as a consequence of the preceding conjecture and the fact that graphs embeddable in a fixed genus have uniformly bounded degeneracy; it is therefore not an independent conjecture in the source.