q,t-positivity conjecture for threshold-graph flow polytopes

Let $G$ be a threshold graph with $n+1$ vertices. The weighted Ehrhart series $\operatorname{Ehr}_{q,t}({\mathcal{F}}_G(-n,1,\ldots,1))$ is a polynomial with nonnegative coefficients. q,t-positivity conjecture.

$$\operatorname{Ehr}_{q,t}({\mathcal{F}}_G(-n,1,\ldots,1))\in\mathbf{N}[q,t].$$

By Haglund's result, the complete-graph case is the bigraded Hilbert series of diagonal harmonics, while other graphs can have negative coefficients. The conjecture has been verified up to $n=9$.