The persistence conjecture for first local cohomology of edge ideals

Let ${\Gamma}$ be a connected graph, let $R$ be the polynomial ring associated to its vertices, let $I$ be the edge ideal of ${\Gamma}$, and let ${\mathfrak m}$ be the homogeneous maximal ideal. Persistence conjecture. If

$$H_{\mathfrak m}^1(R/I^t) \neq 0$$

for some $t>0$, then

$$H_{\mathfrak m}^1(R/I^{t+1}) \neq 0.$$

The preceding theorem establishes the analogous assertion for symbolic powers; the ordinary-power version is posed because the authors are unable to find a counterexample.