Halperin's conjecture on negative-degree derivations of positively elliptic algebras

Let

$$H^* = \mathbb{Q}[x_1,\ldots,x_k]/(u_1,\ldots,u_k)$$

be an Artinian complete intersection algebra with grading concentrated in even degrees; such algebras are called positively elliptic algebras. A derivation of degree $|\delta|\in\mathbb{Z}$ is a linear map $\delta:H^*\to H^*$ that increases degree by $|\delta|$ and satisfies

$$\delta(xy)=\delta(x)y+(-1)^{|\delta||x|}x\delta(y)$$

for homogeneous elements $x,y$.

Halperin's conjecture. If $H^*$ is a positively elliptic algebra, then $H^*$ does not admit a non-trivial derivation of negative degree.

This is Meier's algebraic reformulation of Halperin's 1976 conjecture that the Serre spectral sequence of every fibration with positively elliptic fiber degenerates. The conjecture has been confirmed in formal dimensions up to $20$, while the general case remains open.