Nasr's monotonicity conjecture for Kazhdan–Lusztig and $Z$-polynomials

Let $M$ and $M'$ be matroids, and let $\varphi:M\to M'$ be a morphism in the category $\mathcal{M}$. Let $P_M(t)$ and $P_{M'}(t)$ denote their Kazhdan–Lusztig polynomials, and let $Z_M(t)$ and $Z_{M'}(t)$ denote their $Z$-polynomials. Nasr's monotonicity conjecture. For any morphism $\varphi:M\to M'$ in $\mathcal{M}$, the polynomials

$$P_M(t)-P_{M'}(t)\qquad\text{and}\qquad Z_M(t)-Z_{M'}(t)$$

have non-negative coefficients. This conjecture extends an unpublished conjecture of Gedeon for uniform matroids and predicts that these valuative invariants are monotone under matroid morphisms, as is known for the Chow and augmented Chow polynomials.