Spin torus-action conjecture for simply connected manifolds

Let $n\geq 1$, and let $M$ be a closed, spin, simply connected $(n+2)$-manifold with an effective $T^n$-action.

Spin torus-action conjecture. The manifold $M$ is homeomorphic to one of $S^3$, $S^4$, $S^5$, or

$$\#^{n}_{i=2}m_i\cdot S^i\times S^{n+2-i}.$$

This is a purely topological classification statement underlying the proposed refinement of the domain-of-outer-communication conjecture. The supplied context says that the refined conjecture is verified in the relevant dimensions $n=2,3,4$ under the stated spin condition, but does not establish the claim for all $n\geq 1$.