The inner radius of nodal domains in high dimensions

We prove that every nodal domain of an eigenfunction of the Laplacian of eigenvalue $\lambda$ on a $d$-dimensional closed Riemannian manifold contains a ball of radius $c\lambda^{-1/2}(\log\lambda)^{-(d-2)/2}$. This ball is centered at a point at which the eigenfunction attains its maximum in absolute value within the nodal domain.
Comment: 15 pages. v2: Presentation of proof of main theorem significantly simplified following Sasha Logunov's suggestion