The indecomposability conjecture for flag Gorenstein* complexes

Let $K$ be a flag $n$-dimensional Gorenstein* complex with $n\geq 3$. The quotient of its reduced cohomology ring by the top-dimensional class is

$$\widetilde H^*(\mathcal {Z}_{K})/([\mathcal {Z}_{K}]).$$

Indecomposability conjecture. This graded ring is indecomposable, and consequently $\mathcal {Z}_{K}$ is a prime manifold. The analogous assertion is proved in the paper for flag $2$-spheres; the higher-dimensional case is presented as an open extension, with no counterexample known.