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

Let $\mathcal{L}_{P\times[\ell]}(R^q)$ denote the set of labelings with the stated restriction function, and let $\xi$ be the statistic defined in the preceding formulation. Translated triple promotion homomesy conjecture. The triple

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

is $4$-mesic. This is another equivalent formulation of the antipodal symmetry conjecture and remains open.