Characterization of locally standard, $\mathbb{Z}$-equivariantly formal manifolds in general position

We give a characterization of locally standard, $\mathbb{Z}$-equivariantly formal manifolds in general position. In particular, we show that for dimension $2n$ at least $10$, to every such manifold with labeled GKM graph $\Gamma$ there is an equivariantly formal torus manifold such that the restriction of the $T^n$-action to a certain $T^{n-1}$-action yields the same labeled graph $\Gamma$, thus showing that the (equivariant) cohomology with $\mathbb{Z}$-coefficients of those manifolds has the same description as that of equivariantly formal torus manifolds.