  We propose a combinatorial approach to the following strengthening of Gal's
conjecture: $\gamma(\Delta)\ge \gamma(E)$ coefficientwise, where $\Delta$ is a
flag homology sphere and $E\subseteq \Delta$ an induced homology sphere of
codimension $1$. We provide partial evidence in favor of this approach, and
prove a nontrivial nonlinear inequality that follows from the above conjecture,
for boundary complexes of flag $d$-polytopes: $h_1(\Delta) h_i(\Delta) \ge
(d-i+1)h_{i-1}(\Delta) + (i+1) h_{i+1}(\Delta)$ for all $0\le i\le d$.
