The flag upper bound conjecture for odd-dimensional spheres

Let $J_m(n)$ be the simplicial $(2m-1)$-sphere on $n$ vertices obtained as the join of $m$ cycles, each having either $\lfloor n/m\rfloor$ or $\lceil n/m\rceil$ vertices. Flag upper bound conjecture. If $\Delta$ is a flag $(2m-1)$-sphere with $n$ vertices, then

$$f_i(\Delta)\leq f_i(J_m(n))$$

for all $1\leq i\leq 2m-1$. Furthermore, if equality holds for some $i$, then $\Delta=J_m(n)$. The source discusses this as the flag analogue of the upper bound theorem; it does not give a general resolution.