Where are the N-Koszul algebras of finite global dimension?

The class of $N$-Koszul graded algebras of finite global dimension has gained lots of attention in recent years, especially in the study of Artin-Schelter regular algebras. While structurally rich and concrete, the only known examples of such algebras are either when $N=2$, i.e. the algebra is Koszul, or when $N=3$. Under a mild Hilbert series assumption, we rule out the existence of $N$-Koszul graded algebras of finite global dimension for $N$ not prime. Furthermore, we establish strong restrictions on the global dimension of such algebras. This suggests that perhaps the existence of 3-Koszul algebras with finite global dimension and `nice’ Hilbert series is an anomaly.