Sufficiency of two conditions for long corners is unclear

Determine whether, for ℓ≥3, box-seminorm control of a sequence a:ℤ→ℤ on all systems and equidistribution of a on nilsystems suffice to imply the long-corners weak-limit identity lim_{N→∞} (1/N)∑_{n=1}^N ∫ f_0·T_1^{a(n)}f_1⋯T_ℓ^{a(n)}f_ℓ dμ = lim_{N→∞} (1/N)∑_{n=1}^N ∫ f_0·T_1^{n}f_1⋯T_ℓ^{n}f_ℓ dμ for all f_0,...,f_ℓ.

Background

A degree-lowering argument yields a two-condition criterion (box-seminorm control plus nilsystem equidistribution) for weak-limit identities in 2-corners.

Whether the same two conditions suffice for longer corners is presently unresolved.