Flag upper bound conjecture for Eulerian complexes

Let $m\geq 2$, and let $\Delta$ be a flag $(2m-1)$-dimensional complex on $n$ vertices. Assume that $\Delta$ is Eulerian. For $m\geq 1$, let $J_m(n)$ be the join of $m$ cycles with lengths as equal as possible and total vertex number $n$.

Flag Eulerian upper bound conjecture. For every $1\leq i\leq 2m-1$,

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

The paper proves this in the case $m=2$ and for flag 5-manifolds when $i=1$. The general extension from flag homology spheres to flag Eulerian complexes remains open.