Slowest-chain conjecture for the Miura-ori flip graph

Let $m\ge n$. A $\nu$-chain is a sequence of integers arising from the successive sizes of a monotone chain of order ideals in the $m\times n$ grid, where $\nu$ is the associated size-growth function; the antidiagonal chain is the chain obtained by adding cells along successive antidiagonals. Let $D(m,n)$ denote the value attained by the antidiagonal chain.

Slowest-chain conjecture. Among all $\nu$-chains, the antidiagonal chain maximises

$$\sum_\ell\min(c_\ell,\,mn-c_\ell),$$

with maximum value $D(m,n)$.

This conjecture would imply that the antidiagonal chain gives the largest possible displacement and hence that the diameter of the Miura-ori flip graph equals $D(m,n)$. Exhaustive enumeration confirms it through $5\times4$; a general proof remains open.