Donovan–Wemyss conjecture for contraction algebras

Let $\mathcal{X}_1\to\operatorname{Spec}\mathcal{R}_1$ and $\mathcal{X}_2\to\operatorname{Spec}\mathcal{R}_2$ be crepant contractions as in the two situations described above, with $\mathcal{X}_i$ smooth and $\mathcal{R}_i$ complete local cDV singularities. Let $\mathrm{A}_{\mathrm{con}}$ and $\mathrm{B}_{\mathrm{con}}$ be their contraction algebras, and assume that each contraction has only one curve.

Donovan–Wemyss conjecture. The base singularities are isomorphic if and only if the contraction algebras are isomorphic:

$$\mathcal{R}_1\cong\mathcal{R}_2\iff \mathrm{A}_{\mathrm{con}}\cong\mathrm{B}_{\mathrm{con}}.$$

This conjecture asserts that, for smooth crepant contractible curve neighbourhoods, the contraction algebra is a complete invariant of the corresponding singularity. The divisor-to-curve case remains open.