Extremal gamma-vector conjecture for even-dimensional flag spheres

Let $d\geq 4$ be even, and let $\Delta$ be a flag simplicial $(d-1)$-sphere on $n$ vertices. Let $f_{i-1}(r,m)$ denote the number of $i$-cliques in the complete $r$-partite Turán graph on $m$ vertices with parts as equal in size as possible. Extremal gamma-vector conjecture. For some $2\leq i\leq d/2$, the equality

$$\gamma_i(\Delta)=f_{i-1}\left(\frac{d}{2},n-2d\right)$$

is equivalent to $\Delta$ being the join of $d/2$ cycles whose lengths are as equal as possible. This is an equality characterization for the conjectured upper bounds on the $\gamma$-vector of flag spheres. The supplied status evidence says that the case $d=4$ was confirmed when $\gamma_1$ is sufficiently large, while the full statement remains unresolved.