  For triangulations generated by the adaptive bisection algorithm by Maubach
and Traxler we prove existence of a regularized mesh function with grading two.
This sharpens previous results in two dimensions for the newest vertex
bisection and generalizes them to arbitrary dimensions. In combination with
Diening et al. (2021) this yields $H^1$-stability of the $L^2$-projection onto
Lagrange finite element spaces for all polynomial degrees and dimensions
smaller than seven.
