Nevo's cone containment conjecture for flag and balanced spheres

Let $H_{\mathrm{bal}}$ be the cone generated by the $h$-vectors of balanced $d$-polytopes or balanced homology $(d-1)$-spheres, and let $H_{\mathrm{flag}}$ be the cone generated by the $h$-vectors of flag $d$-polytopes or flag homology $(d-1)$-spheres. Nevo's cone containment conjecture.

$$H_{\mathrm{flag}}\subseteq H_{\mathrm{bal}}.$$

The conjecture is presented as completely open, even in the polytopal case.