Degenerate tiling incidence conjecture over division rings

Let $\mathcal T$ be a simple tiling of a surface of positive genus, meaning that its underlying graph is $(n+1)$-degenerate, and let ${\mathrm D}$ be a division ring. The incidence theorem corresponding to $\mathcal T$ is an incidence statement in projective space $\mathbb{P}^n({\mathrm D})$. Degenerate tiling incidence conjecture. The incidence theorem corresponding to $\mathcal T$ holds in $\mathbb{P}^n({\mathrm D})$ if and only if ${\mathrm D}$ is commutative. This would extend the preceding argument from the specific tiling considered there to all simple $(n+1)$-degenerate tilings of positive-genus surfaces; combined with bounds on the degeneracy of graphs embeddable in a fixed genus, it would imply corresponding dimension thresholds for all simple tilings of a fixed positive genus.