Gao's nonnegative pattern-expansion conjecture for principal specializations

Let $S_n$) denote the permutations of ${1,2,\ldots,n}$, with $S_0=\{\emptyset\}$. For a permutation $w$, let $\Upsilon_w$ be its principal specialization, let $p_v(w)$ count occurrences of the permutation pattern $v$ in $w$, and define $c_\emptyset=1$ and

$$c_w:=\Upsilon_w-\sum_{\substack{v\in S_m\\0\leq m<n}}c_v p_v(w).$$

Gao's conjecture. For any permutation $w$, $c_w\geq 0$. Equivalently,

$$\Upsilon_w=\sum_v c_v p_v(w),\qquad c_v\in\mathbb{Z}_{\geq 0}.$$

This would imply previously established lower bounds for principal specializations of Grothendieck polynomials; its status is not resolved in the supplied text.