The restricted bipartite connectivity conjecture for stable symbolic-power depth

Let $G$ be a simple graph. Write $S$ for the polynomial ring associated to $G$, let $I(G)$ be its edge ideal, and let $\operatorname{bc}'(G)$ denote the restricted bipartite connectivity number defined by minimizing the number of restricted connected components over maximal bipartite subgraphs of $G$.

Restricted bipartite connectivity conjecture. The stable value of the depth of the symbolic powers of $I(G)$ is

$$\lim_{s \to \infty} \operatorname{depth} S/I(G)^{(s)} = \operatorname{bc}'(G).$$

The previously established upper bound uses the ordinary bipartite connectivity number, and examples show that this bound can be strict. The restricted invariant is introduced to give the conjecturally exact stable depth, but the source does not provide a resolution.