The cohomology-gap conjecture for elliptic minimal algebras

Let $(\Lambda V,d)$ be an elliptic minimal algebra whose differential has homogeneous length. Write

$$e=\operatorname{cat}_0(\Lambda V)=\operatorname{e}_0(\Lambda V),$$

where $e$ is given by the formula of the theorem on homogeneous-length differentials and gaps in the Toomer invariant. Cohomology-gap conjecture. Either

$$\dim\bigl(H^*_k(\Lambda V)\bigr)\geq 2 \quad\text{for each } k=1,\ldots,e-1,$$

or $H^*(\Lambda V)$ is a truncated polynomial algebra on a single generator. This proposes that the conclusion of the preceding theorem holds beyond the cases covered by its hypotheses, except for the single-generator truncated-polynomial case.