Vershik's conjecture on finite dimensionality of generic quadratic algebras

Let $n\in{\mathbb N}$ with $n\geqslant 3$. Consider a generic quadratic ${\mathbb C}$-algebra, meaning a quadratic algebra in $n$ generators whose defining relations are generic in the Zariski sense, with $n(n-1)/2$ relations. Vershik's conjecture. Such an algebra is finite dimensional. This addresses the point $d=n(n-1)/2$ in the difficult interval $n^2/4<d<n^2/2$, and would establish finite dimensionality for a significant family of generic quadratic algebras. The source presents it as an open conjecture.