Positivity conjecture for Newton-polytope support coefficients

For $w\in S_n$, write

$$\theta_w=1+\sum_u d_u\,p_u(w),$$

where $\theta_w$ is the number of supports of $\mathfrak{S}_w(x)$ and $p_u(w)$ counts occurrences of the permutation pattern $u$ in $w$. The coefficients are computed recursively by

$$d_u=\theta_u-1-\sum_{\sigma\in S_\ell\atop \ell<m}d_\sigma\,p_\sigma(u),$$

with the same stability property $d_u=0$ when the final entry of $u\in S_m$ is $m$. Support-coefficient positivity conjecture. For any permutation $u$, $d_u\in\mathbb{Z}_{\geq 0}$. This is presented as an analogue of Gao's conjecture; no general resolution is given.