Asymptotic Stanley-depth conjecture for symbolic powers of cover ideals

Let $G$ be a graph with $n$ vertices, let $J(G)$ be its cover ideal, and let $J(G)^{(k)}$ denote its $k$-th symbolic power. Let $\nu_o(G)$ denote the parameter used in the source for the symbolic-power depth formula. Cover-ideal symbolic-power conjecture. One has

$$\lim_{k\rightarrow\infty}{\rm sdepth}(S/J(G)^{(k)})=n-\nu_o(G)-1$$

and

$$\lim_{k\rightarrow\infty}{\rm depth}(J(G)^{(k)})=n-\nu_o(G).$$

The source establishes corresponding eventual Stanley-inequality bounds for cover ideals, but leaves these exact limiting values as a conjecture.