The Realisation Problem for contraction algebras

Let $f\in\mathbb{C}\langle\!\langle x,y\rangle\!\rangle$, and define its Jacobi algebra by

$$\EuScript{J}\mathrm{ac}(f)=\frac{\mathbb{C}\langle\!\langle x,y\rangle\!\rangle}{(\!(\updelta_x f,\updelta_y f)\!)}.$$

Here $\updelta_x$ and $\updelta_y$ are the cyclic derivatives, and $(\!(I)\!)$ denotes the closure of the two-sided ideal generated by $I$.

Realisation Problem. If $f\in\mathbb{C}\langle\!\langle x,y\rangle\!\rangle$ satisfies $\operatorname{GKdim}\EuScript{J}\mathrm{ac}(f)\leq 1$, then $\EuScript{J}\mathrm{ac}(f)$ is geometric. In particular, every finite-dimensional superpotential algebra

$$\frac{\mathbb{C}\langle\!\langle x,y\rangle\!\rangle}{(\!(\updelta_x f,\updelta_y f)\!)}$$

can be constructed as the contraction algebra of some irreducible $3$-fold flop.

This asks which algebras can occur as contraction algebras of crepant resolutions of cDV singularities. The source presents the statement as covering the one-curve case and notes that more general statements exist but are harder to state; no resolution status is supplied.