  There has been recent interest in lower bounds for the principal
specializations of Schubert polynomials $\nu_w := \mathfrak S_w(1,\dots,1)$. We
prove a conjecture of Yibo Gao in the setting of $1243$-avoiding permutations
that gives a lower bound for $\nu_w$ in terms of the permutation patterns
contained in $w$. We extended this result to principal specializations of
$\beta$-Grothendieck polynomials $\nu^{(\beta)}_w := \mathfrak
G^{(\beta)}_w(1,\dots,1)$ by restricting to the class of vexillary
$1243$-avoiding permutations. Our methods are bijective, offering a
combinatorial interpretation of the coefficients $c_w$ and $c^{(\beta)}_w$
appearing in these conjectures.
