Geometric Realisation Conjecture for Jacobi algebras

Let $f$ be a noncommutative potential, and let $\operatorname{Jac} f$ denote its Jacobi algebra. Call $f$ geometric if $\operatorname{Jac} f$ is isomorphic to the contraction algebra of some contraction $\mathcal{X}\to\operatorname{Spec}\mathcal{R}$ in either of the two deformation situations described above.

Geometric Realisation Conjecture. Every $f$ whose Jacobi algebra satisfies

$$\dim \operatorname{Jac} f\leq 1$$

is geometric.

Tens of thousands of computer searches have found no counterexamples, but existing methods do not yet seem strong enough to prove the claim. A new ingredient in noncommutative singularity theory may be needed if the conjecture is true.