Representability conjecture for finitely generated PI-algebras with square-zero Jacobson radical

Let $A$ be a finitely generated PI-algebra over the field of rational numbers, and let $J$ be its Jacobson radical. Representability conjecture. If

$$J^2=0,$$

then $A$ is representable. This would identify the square-zero radical case as representable; the surrounding discussion notes that known examples of nonrepresentable finitely generated PI-algebras have Jacobson radical nilpotency degree at least three, but no resolution of this claim is given.