Typical-orbit conjecture for interval-closed sets of products of two chains

Fix a positive integer $m$. Let $r(n)$ be the fraction of interval-closed sets of $[m]\times[n]$ that belong to rowmotion orbits of size $m+n+1$. Typical-orbit conjecture. For every positive integer $m$, one has

$$r(n)\to 1\quad\text{as }n\to\infty.$$

This conjecture predicts that although rowmotion on interval-closed sets can have complicated behavior and large order, almost all interval-closed sets lie in orbits of the indicated size as $n$ grows. The source presents this as supported by experimental evidence and outlines a possible proof strategy, but does not establish it.