The depth-drop conjecture for powers of edge ideals

Let ${\Gamma}$ be a connected non-bipartite graph, let $R$ be the polynomial ring associated to its vertices, and let $I$ be the edge ideal of ${\Gamma}$. Depth-drop conjecture. If

$$\operatorname{depth} R/I^t = 1,$$

then

$$\operatorname{depth} R/I^{t+3} = 0.$$

The preceding theorem proves a related result under the stronger hypothesis that a suitable localized quotient has depth zero. The stated assertion is posed as an open problem because the authors are unable to find a counterexample.