Gal's stability conjecture for dense flag homology manifolds

Let $m\geq 2$, and let $\Delta$ be a flag homology $(2m-1)$-manifold with $n$ vertices. The Gal's stability conjecture. There exists a constant $b=b(m)$ such that if

$$f_{2m-1}(\Delta)>f_{2m-1}(J_m(n))-b(m)n^{m-1},$$

then $\Delta$ is the join of $m$ cycles. This proposes a higher-dimensional stability analogue of Gal's conjecture: manifolds with facet number close to the conjectured maximum should have the corresponding join-of-cycles structure. The source notes that the three-dimensional case holds asymptotically, but gives no resolution of this generalization.