  Let $(X,\mu)$ be a probability space, $G$ a countable amenable group and
$(F_n)_n$ a left F\o lner sequence in $G$. This paper analyzes the
non-conventional ergodic averages \[\frac{1}{|F_n|}\sum_{g \in
F_n}\prod_{i=1}^d (f_i\circ T_1^g\cdots T_i^g)\] associated to a commuting
tuple of $\mu$-preserving actions $T_1$, ..., $T_d:G\curvearrowright X$ and
$f_1$, ..., $f_d \in L^\infty(\mu)$. We prove that these averages always
converge in $\|\cdot\|_2$, and that they witness a multiple recurrence
phenomenon when $f_1 = \ldots = f_d = 1_A$ for a non-negligible set $A\subseteq
X$. This proves a conjecture of Bergelson, McCutcheon and Zhang. The proof
relies on an adaptation from earlier works of the machinery of sated
extensions.
