Centered-set conjecture for k-chains in grid posets

Let $[m]^d$ be the product poset, ordered coordinatewise: $\mathbf{a}\leq\mathbf{b}$ when $a_i\leq b_i$ for every $i\in[d]$. A $k$-chain is a set of $k$ distinct points satisfying $\mathbf{a}_1\leq\cdots\leq\mathbf{a}_k$. For $\mathcal{F}\subseteq[m]^d$, call $\mathcal{F}$ $m$-centered if, whenever $\mathbf{a}\in\mathcal{F}$ and $\mathbf{b}\notin\mathcal{F}$,

$$\left\lvert\sum_{i=1}^d a_i-\frac{dm}{2}\right\rvert\leq\left\lvert\sum_{i=1}^d b_i-\frac{dm}{2}\right\rvert,$$

and equality implies $\sum_{i=1}^d a_i\geq\sum_{i=1}^d b_i$.

Centered-set conjecture. Given $m$, there exists a number $d_0(m)$ such that if $d\geq d_0(m)$, then the sets minimizing the number of $k$-chains among subsets of $[m]^d$ of any prescribed size are $m$-centered.

This is a proposed extension of Kleitman's question from the Boolean lattice to grid posets. The source emphasizes that the conjecture is a natural guess with little supporting evidence and does not claim uniqueness of the minimizers.