Furstenberg-Johnson-Lesigne-Wierdl conjecture on mean convergence along arithmetic progressions

Let $a\colon {\mathbb N}\to {\mathbb Z}$ be a sequence and let $\ell\in {\mathbb N}$. A sequence is good for mean convergence along $\ell$-term arithmetic progressions if the corresponding multiple ergodic averages converge in $L^2(\mu)$ for every measure-preserving system. It is good for $\ell$-step equidistribution when it satisfies the paper's $\ell$-step equidistribution property. Furstenberg-Johnson-Lesigne-Wierdl conjecture. The following properties are equivalent: $a$ is good for mean convergence along $\ell$-term arithmetic progressions for all systems, and $a$ is good for $\ell$-step equidistribution. This would characterize mean convergence along arithmetic progressions through an equidistribution condition, extending the classical one-term criterion; the source gives no resolution of the equivalence.