Elizalde–Plante–Roby–Sagan mesicity conjecture for fence rowmotion

Let $F(\beta)$ be a fence and let $L(\beta)$ be its lattice of lower order ideals. Suppose $k\ge 2$ and $\beta=(k-1,k,k,\ldots,k,k-1)$ has an odd number of parts. Under rowmotion on $L(\beta)$, define the statistic $\operatorname{st}:L(\beta)\to\{0,1,2,\ldots\}$ by $\operatorname{st}(I)=\#I$. A statistic is $c$-mesic if its average over every rowmotion orbit is $c$. Elizalde–Plante–Roby–Sagan's mesicity conjecture. The statistic $\operatorname{st}$ is $n/2$-mesic, where $n=\#F(\beta)$. The conjecture predicts uniform orbit averages for the cardinality statistic under rowmotion on this family of fences; the source says that it remained unresolved by the cited authors.