Lupton's cohomology-dimension conjecture for elliptic Sullivan algebras

Let $(\Lambda V,d)$ be an elliptic Sullivan algebra with homogeneous differential of length $l$. Write

$$e=e_0(\Lambda V,d)=\dim V^{\operatorname{odd}}+(l-2)\dim V^{\operatorname{even}}.$$

Lupton's conjecture. Either

$$\dim H_k^*(\Lambda V,d)\geq 2\quad\text{for }k=1,\ldots,e-1,$$

or $H^*(\Lambda V,d)$ is a truncated polynomial algebra on a single generator.

Lupton established the conjecture partially under the additional hypothesis that $\ker(d:V^{\operatorname{odd}}\to\Lambda V)$ is non-zero. The paper supplies a complementary result for coformal elliptic Sullivan algebras with differential of length $2$, leaving the general case open.