Monotonicity conjecture for the projective-dimension–regularity spectrum

Let $n\geq 2$ and define

$$\operatorname{pdreg}(n)=\{(p,r)\mid \text{there is a graph }G\text{ over }n\text{ vertices with }\operatorname{pd}(G)=p\text{ and }\operatorname{reg}(G)=r\}.$$

Monotonicity conjecture. Let $r,n\geq 2$ be arbitrary integers. If $(p,r)\in\operatorname{pdreg}(n)$, then $(p,r-1)\in\operatorname{pdreg}(n)$. The preceding results show that the spectrum contains broad ranges of projective dimensions for fixed regularity, and computations suggest that these ranges become smaller as regularity increases. The conjecture asserts the corresponding monotonicity across successive regularity values.