Furstenberg-Lesigne-Wierdl recurrence conjecture for arithmetic progressions

Let $a\colon {\mathbb N}\to {\mathbb Z}$ be a sequence. A sequence is good for $\ell$-step irrational equidistribution if it satisfies the corresponding irrational equidistribution property, and it has good divisibility properties in the sense used by the source. Furstenberg-Lesigne-Wierdl recurrence conjecture. If $a$ is good for $\ell$-step irrational equidistribution and has good divisibility properties, then it is good for recurrence along $\ell$-term arithmetic progressions. This generalizes the known one-term Kamae-Mendès France criterion; the source gives no resolution of the higher-order assertion.