Equidistribution conjecture for the alpha, beta and gamma statistics

For Gog, Magog and GOGAm triangles, define the statistics $\alpha$, $\beta$, and $\gamma$ as in the paper; in particular, $\alpha_{\mathrm{Gog}}(X)$ counts indices $k$ with $X_{k,1}=1$, $\beta_{\mathrm{Gog}}(X)=X_{1,1}$, and the corresponding Magog and GOGAm statistics are defined using the Schützenberger involution and the stated boundary data. Alpha–beta–gamma equidistribution conjecture. For any $n$, the three statistics $\alpha$, $\beta$, and $\gamma$ are equienumerated on, respectively, Gog, Magog, and GOGAm triangles. This proposes a refined enumerative correspondence beyond the basic equality of cardinalities; the source gives no resolution.