Linear obstruction conjecture for regularity–$h$-degree pairs of connected graphs

Let $r,d\in\mathbb{Z}_{\ge 0}$, and let $G$ be a connected graph. Write $R$ for the polynomial ring associated with $G$, $J(G)$ for its cover ideal, $\operatorname{reg}(R/J(G))$ for the Castelnuovo–Mumford regularity, and $h_{R/J(G)}(t)$ for the numerator of the Hilbert series of $R/J(G)$. Linear obstruction conjecture. If either

$$r\leq\left\lceil\frac{d}{2}\right\rceil$$

or

$$d\geq\left\lceil\frac{2r-1}{3}\right\rceil,$$

then there is no connected graph $G$ such that

$$\bigl(\operatorname{reg}(R/J(G)),\ \deg h_{R/J(G)}(t)\bigr)=(r,d).$$

The claim extends the preceding obstruction that no connected graph realizes $(1,d)$ for $d\geq 2$. It is motivated by computations suggesting that unrealizable pairs are constrained by simple linear inequalities; the supplied source gives no resolution, so the conjecture remains open.