Classification for joint ergodicity over ℤ^ℓ-polynomial actions

Prove or disprove that for polynomials a_1,...,a_ℓ∈ℤ^ℓ[t] and a system (X,𝔛,μ,T_1,...,T_ℓ), the sequence (T^{a_1(n)},...,T^{a_ℓ(n)})_n is jointly ergodic if and only if (i) (T^{a_1(n)}×⋯×T^{a_ℓ(n)})_n is ergodic on (X^ℓ,𝔛^{⊗ℓ},μ^ℓ) and (ii) (T^{a_i(n)}T_j^{-a_j(n)})_n is ergodic on (X,𝔛,μ) for all i≠j.

Background

The survey proves a complete classification for integer polynomials in one variable and poses the higher-dimensional analogue over ℤ^ℓ.

This generalization would unify product and difference ergodicity criteria with joint ergodicity for polynomial actions of ℤ^ℓ.