Noel–Scott–Sudakov–Balogh–Wagner centeredness conjecture for product posets

Let $k$ be a positive integer, and let $n_0$ be an integer. Equip the product poset $\{0,1,\ldots,k\}^n$ with the coordinatewise order. A family is centered when its elements are chosen as close as possible to the middle rank, and the poset has the centeredness property if, for every family size $M$, some centered family of size $M$ minimizes the number of comparable pairs. Noel–Scott–Sudakov–Balogh–Wagner conjecture. For every $k$ there exists an $n_0$ such that if $n\geq n_0$ then the poset $\{0,1,\ldots,k\}^n$ has the centeredness property. This extends Kleitman's theorem for the Boolean lattice, while a counterexample is known for $n=2$ and $k=16$; the asserted eventual centeredness for each fixed $k$ remains open.