Strong B-rigidity conjecture for flag Gorenstein* complexes

A simplicial sphere $K$ is strongly $B$-rigid if, whenever a simplicial sphere $L$ satisfies

$$H^*(\mathcal {Z}_K)/([\mathcal {Z}_K])\cong H^*(\mathcal {Z}_L)/([\mathcal {Z}_L]),$$

then $K$ and $L$ are combinatorially equivalent. Strong B-rigidity conjecture. Every flag $2$-sphere is strongly $B$-rigid; more generally, every flag Gorenstein* complex is strongly $B$-rigid. The statement proposes that the quotient cohomology ring determines the simplicial sphere within this class. The paper does not provide a resolution of the general assertion.