Antipodal rowmotion homomesy conjecture for $[a] \times [2] \times [2]$

Let $[n]=\{1,\ldots,n\}$, let $\mathcal{A}^2(P)$ denote the set of $P$-partitions of height at most $2$, and let $\chi_S$ be the indicator statistic of $S$. For antipodal elements $x,y$ of $[a] \times [b] \times [\ell]$, set $S=\{x,y\}$. Antipodal rowmotion homomesy conjecture. The triple

$$\left(\mathcal{A}^2([a] \times [2] \times [2]),\operatorname*{Row},\chi_S\right)$$

is $2$-mesic. This was verified computationally for $a\leq 6$, but the general statement remains open.