Monotonicity of the symbolic depth function for edge ideals

Determine whether the symbolic depth function k ↦ depth S/J(G)^{(k)} is non-increasing for every edge ideal J(G) of a finite graph G.

Background

The paper contrasts the behavior of symbolic depth functions for several classes of squarefree monomial ideals. Although the symbolic depth function need not be non-increasing for arbitrary squarefree monomial ideals, and the paper establishes analogous non-monotonicity phenomena for complementary edge ideals, the corresponding question for ordinary edge ideals remains unresolved. Here J(G) denotes the edge ideal of a graph G, and S is the associated polynomial ring.