Norm convergence for generalized polynomials (ℓ=2, weakly mixing)

Determine whether for every weakly mixing invertible measure-preserving transformation T and every pair of generalized polynomials a_1,a_2:ℤ→ℤ the double average (1/N)∑_{n=1}^N T^{a_1(n)}f_1·T^{a_2(n)}f_2 converges in L^2(μ) for all f_1,f_2∈L^∞(μ).

Background

The norm convergence problem for generalized polynomial iterates is posed in full generality; the single-term case ℓ=1 follows from the spectral theorem.

Even for the seemingly accessible case ℓ=2 under weak mixing, convergence is not known.