Eastwood–Ezhov's LSA characterization conjecture

Let $\mathcal A$ be a complete abelian left-symmetric algebra (LSA) with a nondegenerate Hessian-type inner product. Let $\operatorname{sDer}(\mathcal A)$ denote its space of infinitesimal similarity derivations. Eastwood–Ezhov's LSA conjecture. If

$$\dim \operatorname{sDer}(\mathcal A)=1,$$

then $\mathcal A$ is the abelian filiform LSA. This is a reformulation of the Cayley-hypersurface characterization conjecture in terms of complete abelian LSAs and their associated polynomial geometry; the source does not state a resolution.