Gao's positivity conjecture for permutation-pattern coefficients

For $w\in S_n$, write

$$\nu_w=1+\sum_{u\in S_m\atop m\leq n}c_u\,p_u(w),$$

where $p_u(w)$ counts occurrences of the pattern $u$ in $w$, and define the coefficients recursively by

$$c_u=\nu_u-1-\sum_{\sigma\in S_\ell\atop \ell<m}c_\sigma\,p_\sigma(u).$$

Here $u$ ranges over permutations and $c_u$ has the stability property $c_u=0$ when the final entry of $u$ is its maximum. Gao's positivity conjecture. For any permutation $u$, $c_u\in\mathbb{Z}_{\geq 0}$. The source records proofs for several pattern-avoidance classes, but not the general assertion.