Positivity conjecture for differences along the threshold-graph poset

Let $\mathcal{P}_n$ be the poset of connected threshold graphs with vertices $0,1,\ldots,n$, ordered by subgraph inclusion: $H\preceq G$ when $H$ is a subgraph of $G$. For threshold graphs $G\succeq H$ in $\mathcal{P}_n$, poset-difference positivity conjecture.

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

The poset is isomorphic to the poset of shifted Young diagrams contained in $(n-1,n-2,\ldots,0)$. Computations up to $n=9$ suggest positivity along its cover relations; the conjecture as stated concerns all comparable threshold graphs.