Host–Kra versus Weyl complexity (conjectured equivalence)

Prove that for any family of essentially distinct integer polynomials a_1,...,a_ℓ, the Host–Kra complexity equals the Weyl complexity, i.e., the minimal Host–Kra factor controlling the averages on totally ergodic systems coincides with that determined via equidistribution on Weyl systems.

Background

The survey defines Host–Kra and Weyl complexities and notes many works conjecturing their equality.

Equality is verified in several cases (e.g., algebraic complexity ≤1), but the general statement remains open.