Grothendieck inclusion–exclusion positivity conjecture

Let $w\in S_n$ and let $u$ be a subword of $w$. For a word $v$, write $|v|$ for its number of entries and $\operatorname{perm}(v)$ for the permutation with the same relative order as $v$. Let $\Upsilon_x(\beta)$ denote the principal specialization of the Grothendieck polynomial indexed by a permutation $x$.

Grothendieck inclusion–exclusion positivity conjecture. For every permutation $w\in S_n$ and every subword $u$ of $w$,

$$\sum_{u\leq v\leq w}(-1)^{|w|-|v|}\Upsilon_{\operatorname{perm}(v)}(\beta)\in\mathbb{Z}_{\geq0}[\beta].$$

Taking $u=\emptyset$ gives Dennin's coefficient-positivity conjecture. The claim is proved in the supplied paper when $w$ avoids the $1423$ pattern, but its general status is not resolved there.