The polyhedral-complex degree-bound conjecture for spline dimensions

Let $P$ be a planar polyhedral complex, and let F=\max\{n\mid\text{\rm {there is an n-gon in }}P\}. Let $k$ and $r$ be the parameters in the spline-dimension formula of Theorem~. The polyhedral-complex degree-bound conjecture. Theorem~ applies if $k\geq(F-1)(r+1)-1$. The source presents this as an additional open question after citing a weaker known bound, so determining the sharper threshold would improve the dimension formula for planar polyhedral complexes.