The finite-field quadratic-growth conjecture for simple algebras

Let $k$ be a finite field and let $A$ be a finitely generated $k$-algebra of quadratic growth. Finite-field quadratic-growth conjecture. Then $A$ is not simple. The conjecture is motivated by examples of finitely generated simple Goldie algebras of quadratic growth over fields of characteristic $0$, while analogous constructions over finite fields are obstructed by results showing that many finitely generated prime algebras satisfy a polynomial identity. Its status is unresolved here.