The homeomorphism conjecture for connected sums of moment-angle complexes

Let $K_1$ and $K_2$ be generalized homology spheres of dimension $n-1$, with $m_1$ and $m_2$ vertices, respectively. Let

$$\lambda(i)=\binom{m_1+m_2-2n}{i}-\binom{m_1-n}{i}-\binom{m_2-n}{i}.$$

The operation $\mathcal{G}$ on manifolds is the gyration, and $\mathcal{G}^r$ denotes its $r$-fold iteration. The connected-sum homeomorphism conjecture. One has

$$\mathcal {Z}_{K_1\#K_2}\cong \mathcal {G}^{m_2-n}(\mathcal {Z}_{K_1})\#\mathcal {G}^{m_1-n}(\mathcal {Z}_{K_2}) \#\underset{i=2}{\overset{m_1+m_2-2n}{\sharp}}\lambda(i)(S^{i+1}\times S^{m_1+m_2-i-1}).$$

The preceding theorem identifies the cohomology ring of the left-hand side with the corresponding algebra, and the conjecture asks whether this ring-level decomposition is induced by a homeomorphism of manifolds. The special case in which one complex is a simplex boundary is known, while the general case is left open.