Translated promotion homomesy conjecture for $[a] \times [2] \times [2]$

Let $\mathcal{L}_{P \times [\ell]}(R^q)$ denote the set of labelings subject to the indicated restriction function, let $\xi(f,(x_1,x_2),b)$ be the statistic defined by

$$\xi(f,(x_1,x_2),b)=\sum_{k=1}^b k\cdot\#\{(x_1,x_2,j)\in[a]\times[2]\times[2]:f(x_1,x_2,j)=x_1+x_2+k-1\},$$

and let $\operatorname*{Pro}$ denote promotion. Translated promotion homomesy conjecture. For every $a\in\mathbb{N}$,

$$\left(\mathcal{L}_{([a]\times[2])\times[2]}(R^{a+3}),\operatorname*{Pro},\xi(f,(x_1,x_2),2)+\xi(f,(a+1-x_1,3-x_2),2)\right)$$

is $4$-mesic. It is presented as equivalent to the antipodal rowmotion conjecture and is therefore open, although the original conjecture was verified for $a\leq6$.