Alahmadi–Alsulami–Jain–Zelmanov equivalence conjecture for algebra growth functions

Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be a function. Say that $f$ is equivalent to the growth of an algebra when it is equivalent, in the growth-function sense, to the growth function of that algebra; similarly, distinguish finitely generated algebras, primitive algebras, and nil algebras.

Alahmadi–Alsulami–Jain–Zelmanov conjecture. The following are equivalent:

• $f$ is equivalent to the growth of a finitely generated algebra;
• $f$ is equivalent to the growth of a finitely generated primitive algebra;
• $f$ is equivalent to the growth of a finitely generated nil algebra.

This claim identifies the growth functions realizable by finitely generated algebras with those realizable within the more restrictive classes of primitive and nil algebras. The supplied source gives no resolution status.