Order ideal cardinality homomesy for odd uniform self-dual fences

Let $a$ be a positive integer, let $t$ be odd, and let $F=\breve{F}(a^t)$ be the uniform self-dual fence with $t$ segments. Write $n=\#F$ for its number of elements, and let $\hat{\chi}$ be the order ideal cardinality statistic. Order ideal cardinality conjecture. The statistic $\hat{\chi}$ is $n/2$-mesic under rowmotion.

The source explains that this conjecture is equivalent to a 0-mesy statement involving peak and valley antichain indicators, and records computational confirmation for several parameter ranges.