Flag gamma-positivity conjecture for theta polynomials of homology balls

Let $\Delta$ be a flag homology ball, and suppose that its boundary $\partial\Delta$ is an induced subcomplex of $\Delta$. Let $\theta(\Delta,x)$ denote the polynomial introduced for such homology balls. A polynomial is $\gamma$-positive if it admits an expansion

$$\theta(\Delta,x)=\sum_i\gamma_i x^i(1+x)^{d-2i}$$

with all $\gamma_i\geq 0$. Flag theta-polynomial $\gamma$-positivity conjecture. The polynomial $\theta(\Delta,x)$ is $\gamma$-positive for every flag homology ball $\Delta$ such that $\partial\Delta$ is an induced subcomplex of $\Delta$. This is presented as a flag analogue of the paper's unimodality theorem for theta polynomials. The source does not establish the conjecture in general.