Dwork's specialization conjecture for log-growth Newton polygons

Let $D y=0$ be an ordinary linear $p$-adic differential equation of order $\mu$ with bounded analytic coefficients on the unit disc and a full set of solutions analytic there. Let $\mathrm{NP}_{\log,0}(D)$ be its special log-growth Newton polygon, and let $\mathrm{NP}_{\log,t}(D_t)$ be the generic log-growth Newton polygon obtained from the equation pulled back to the unit disc around a generic point $t$.

Dwork's specialization conjecture. The polygon $\mathrm{NP}_{\log,0}(D)$ is above $\mathrm{NP}_{\log,t}(D_t)$.

This comparison was motivated by Grothendieck's specialization theorem for $F$-isocrystals, conditional on the preceding equality between log-growth and Frobenius Newton polygons. The source's supplied resolution evidence states that the generic case of Dwork's conjecture was proved without any assumption, so the claim is recorded as solved.