Equivalence of the SCC and NSC for flag Gorenstein* complexes

A flag Gorenstein$^*$ $n$-complex $K$ is assumed to satisfy the SCC, and the NSC is the condition that there is no full subcomplex $K_I\subset K$ such that $K_I$ is a suspension and $\widetilde H^{n-1}(K_I)\neq0$. SCC–NSC equivalence conjecture. The SCC and NSC are equivalent for all flag Gorenstein$^*$ complexes. Since the NSC is much easier to verify computationally than the SCC, this conjecture proposes that the necessary NSC condition is also sufficient.