GKM theory and Hamiltonian non-K\"ahler actions in dimension $6$

Using the classification of $6$-dimensional manifolds by Wall, Jupp and \v{Z}ubr, we observe that the diffeomorphism type of simply-connected, compact $6$-dimensional integer GKM $T^2$-manifolds is encoded in their GKM graph. As an application, we show that the $6$-dimensional manifolds on which Tolman and Woodward constructed Hamiltonian, non-K\"ahler $T^2$-actions with finite fixed point set are both diffeomorphic to Eschenburg's twisted flag manifold $SU(3)//T^2$. In particular, they admit a noninvariant K\"ahler structure.