Alternation conjecture for non-negatively curved odd GKM$_3$ manifolds

Let $M^{2n+1}$ be a closed, non-negatively curved odd-dimensional GKM$_3$ manifold admitting an invariant almost contact structure. Its corresponding odd GKM$_3$ graph is the graph associated with $M^{2n+1}$ in odd GKM theory.

Alternation conjecture. The odd GKM$_3$ graph corresponding to $M^{2n+1}$ is alternating.

The preceding argument establishes the analogous conclusion for odd GKM$_4$ manifolds. The conjecture proposes that the alternation property already follows under the weaker GKM$_3$ hypothesis; the supplied text gives no resolution.