We remove the dependence on the `hot-spots' conjecture in two of the main theorems of the recent paper of Nickl (2024, Annals of Statistics). Specifically, we characterise the minimax convergence rates for estimation of the transition operator $P_{f}$ arising from the Neumann Laplacian with diffusion coefficient $f$ on arbitrary convex domains with smooth boundary, and further show that a general Lipschitz stability estimate holds for the inverse map $P_f\mapsto f$ from $H^2\to H^2$ to $L^1$.