The flag upper bound conjecture for even-dimensional spheres

Let $J_m^*(n)$ be the suspension of a join of $m$ balanced cycles as defined in the source, and let $\mathcal{J}_m^*(n)$ be the family of joins $S*C_2*\cdots*C_m$ with $S$ a flag 2-sphere on the prescribed number of vertices. Even-dimensional flag upper bound conjecture. If $\Delta$ is a flag homology $2m$-sphere on $n$ vertices, then

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

for all $i=1,\dots,2m$. If equality holds for some $1\leq i\leq 2m$, then $\Delta\in\mathcal{J}_m^*(n)$. The source says this conjecture is open except for the inequality part when $m=2$.