  We study comparison geometry of manifolds with boundary under a lower
$N$-weighted Ricci curvature bound for $N\in ]-\infty,1]\cup [n,+\infty]$ with
$\varepsilon$-range introduced by Lu-Minguzzi-Ohta. We will conclude splitting
theorems, and also comparison geometric results for inscribed radius, volume
around the boundary, and smallest Dirichlet eigenvalue of the weighted
$p$-Laplacian. Our results interpolate those for $N\in [n,+\infty[$ and
$\varepsilon=1$, and for $N\in ]-\infty,1]$ and $\varepsilon=0$ by the second
named author.
