Comparison of symbolic and ordinary depths of edge ideals

Prove or disprove that, for every edge ideal J(G) and every k ≥ 1, the depth of the k-th symbolic-power quotient is at least the depth of the corresponding ordinary-power quotient, namely depth S/J(G)^{(k)} ≥ depth S/J(G)^k.

Background

The paper identifies a conjectural comparison between symbolic and ordinary powers of edge ideals. The inequality is known for the second and third symbolic powers, but its validity for all powers remains unresolved. This problem concerns arbitrary finite graphs and their edge ideals J(G) in a polynomial ring S.