Flag lower-bound conjecture for edges of simplicial spheres

Fix positive integers $n$ and $m$, and consider flag triangulations of the $(n-1)$-dimensional sphere with $m$ vertices. Let the $(n-2)$-fold suspension of the one-dimensional sphere with $m-2n+4$ vertices be the specified comparison complex. Flag edge lower-bound conjecture. Among all flag triangulations of the $(n-1)$-dimensional sphere with a given number $m$ of vertices, the $(n-2)$-fold suspension over the one-dimensional sphere with $m-2n+4$ vertices has the smallest possible number of edges. This conjecture is presented as equivalent to the nonnegativity of the second $\gamma$-coefficient in Gal's conjecture and concerns a flag analogue of Barnette's Lower Bound Theorem; the source gives no resolution.