Comparison geometry of manifolds with boundary under lower $N$-weighted Ricci curvature bounds with $\varepsilon$-range

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.