Convergence speed for the average density of eigenfunctions for singular Riemannian manifolds

We consider a class of singular Riemannian metrics on a compact Riemannian manifold with boundary and the eigenfunctions of the corresponding Laplace-Beltrami operator. In our setting, the average density of eigenfunctions with eigenvalue less than $\lambda$ converges weakly to the uniform normalised measure on the boundary as $\lambda\to\infty$. In this work, we show a quantitative estimate on the speed of this convergence in the Wasserstein-sense in the transverse coordinate to the boundary.