Dennin's coefficient-positivity conjecture for Grothendieck pattern coefficients

For $w\in S_n$, let $\Upsilon_w(\beta)$ be the principal specialization of the Grothendieck polynomial and define $c_\emptyset(\beta)=1$ and

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

Here $p_v(w)$ counts occurrences of the permutation pattern $v$ in $w$.

Dennin's conjecture. For every permutation $w$,

$$c_w(\beta)\in\mathbb{Z}_{\geq0}[\beta].$$

Equivalently,

$$\Upsilon_w(\beta)=\sum_v c_v(\beta)p_v(w),\qquad c_v(\beta)\in\mathbb{Z}_{\geq0}[\beta].$$

This is presented as a Grothendieck analogue of Gao's conjecture. The supplied text does not state a general resolution; it records a proof for permutations avoiding the $1423$ pattern.