  We prove a $q$-refined tropical correspondence theorem for higher genus
descendant logarithmic Gromov--Witten invariants with a $\lambda_g$ class in
toric surfaces. Specifically, a generating series of such logarithmic
Gromov--Witten invariants agrees with a $q$-refined count of rational tropical
curves satisfying higher valency conditions. As a corollary, we obtain a
geometric proof of the deformation invariance of this tropical count. In
particular, our results give an algebro--geometric meaning to the tropical
count defined by Blechman and Shustin. Our strategy is to use the logarithmic
degeneration formula, and the key new technique is to reduce to computing
integrals against double ramification cycles and connect these integrals to the
non--commutative KdV hierarchy.
