Conjecture on regularity of symbolic and ordinary powers of edge ideals

Let $G$ be a simple graph, and let $I(G)$ be its edge ideal. For each integer $s\geq 1$, the symbolic power $I(G)^{(s)}$ and ordinary power $I(G)^s$ are ideals in the associated polynomial ring, with Castelnuovo–Mumford regularity denoted by $\operatorname{reg}$. Regularity equality conjecture. For all $s\geq 1$,

$$\operatorname{reg}(I(G)^{(s)})=\operatorname{reg}(I(G)^s).$$

The equality would show that the regularity of symbolic powers of edge ideals is governed by the same function as the regularity of ordinary powers, resolving the question of whether symbolic-power regularity is asymptotically linear in this setting. The equality is proved in the paper for $s=2,3$, while the assertion for all $s\geq 1$ remains open.