The single valuation conjecture for homological dimension zero

Let $\mathcal{O}_2=k[x_1,x_2]$ and let $\mathcal{I}\triangleleft\mathcal{O}_2$ be a finite-codimensional, integrally closed ideal. Let $\mathcal{O}$ denote the ambient module in the inclusion $\mathcal{I}\hookrightarrow\mathcal{O}$, and let $\operatorname{homdim}(\mathcal{I}\hookrightarrow\mathcal{O})$ be the lattice homological dimension. Single valuation conjecture. If

$$\operatorname{homdim}(\mathcal{I}\hookrightarrow\mathcal{O})=0,$$

then there exist a discrete valuation $\mathfrak{v}$ and an integer $d_{\mathcal{I}}$ such that

$$\mathcal{I}=\{f\in\mathcal{O}_2:\mathfrak{v}(f)\geq d_{\mathcal{I}}\}.$$

The conjecture is motivated by computations in which every finite-codimensional integrally closed ideal of homological dimension zero was realized by a single valuation. The source gives no resolution.