The single valuation conjecture for integrally closed ideals in two variables

Let $k[x_1,x_2]$ be the polynomial ring in two variables, and let $\mathcal{I}\triangleleft k[x_1,x_2]$ be a finite-codimensional, integrally closed ideal. Write $\mathbb{SH}_{\geq 1}(\mathcal{I}\triangleleft k[x_1,x_2])$ for its reduced lattice homology in degrees at least $1$. Single valuation conjecture. If

$$\mathbb{SH}_{\geq 1}(\mathcal{I}\triangleleft k[x_1,x_2])=0,$$

then $\mathcal{I}$ can be realized by a single (semi-)valuation. This predicts that vanishing of the positive-degree reduced lattice homology detects one-valuation realizability for finite-codimensional integrally closed ideals in two variables. The source gives no resolution of the conjecture.