Link Conjecture for flag homology spheres

Let $\Delta$ be a flag homology sphere and let $v$ be a vertex. Its vertex link is $\operatorname{lk}_v(\Delta)$. Link Conjecture. The link's $\gamma$-vector is coefficientwise at most that of $\Delta$:

$$\gamma(\operatorname{lk}_v(\Delta))\leq\gamma(\Delta).$$

This conjecture immediately implies Gal's conjecture because vertex links in flag homology spheres are again flag homology spheres. The supplied text gives no resolution evidence.