Lutz–Nevo upper bound conjecture for flag spheres

Let $d=2k\geq 4$, and let $\Delta$ be a flag homology $(d-1)$-sphere on $n$ vertices. Let $\mathcal{J}_k(n)$ be the flag sphere obtained by joining $k$ graph cycles, each having either $\lfloor n/k\rfloor$ or $\lceil n/k\rceil$ vertices. Lutz–Nevo upper bound conjecture.

$$\gamma_i(\Delta)\leq\gamma_i(\mathcal{J}_k(n)).$$

Moreover, equality for some $2\leq i\leq k$ holds if and only if $\Delta=\mathcal{J}_k(n)$. The supplied text gives no resolution.