Positselski's conjecture on Koszul algebras of finite global dimension

Let $A$ be a Koszul algebra of finite global homological dimension $d$. Dually, let $B$ be a Koszul algebra with $B_{d+1}=0$ and $B_d\ne 0$.

Positselski's conjecture. Every such algebra $A$ satisfies

$$\dim A_1\ge d,$$

and, dually, every such algebra $B$ satisfies

$$\dim B_1\ge d.$$

The conjecture was recorded in Positselski's monograph and is refuted by an algebra of Iyudu and Shkarin, which the paper studies using the Anick resolution.