The Realisation Conjecture for Jacobi algebras of noncommutative functions

Let $f\in\mathbb C\langle\kern -2.5pt\langle \mathsf{x}\rangle\kern -2.5pt\rangle$ be a noncommutative formal power series, 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 $\mathcal{X}\to\operatorname{Spec}\mathcal{R}$ of the type described in the paper.

The Realisation Conjecture. Every $f\in\mathbb C\langle\kern -2.5pt\langle \mathsf{x}\rangle\kern -2.5pt\rangle$ whose Jacobi algebra satisfies

$$\operatorname{Jdim}\operatorname{Jac}(f)\leq 1$$

is geometric.

This conjecture proposes that the low-dimensional Jacobi algebras arising from noncommutative formal functions are all realised geometrically as contraction algebras. The source reports the conjecture as open; in particular, one unresolved case is when $f\cong x^3+\mathcal{O}_4$, which requires further analysis.