Background: Let $U\subset \mathbb{R}^n$ be a bounded convex domain. By classical results, the (signed) Laplacian $-\Delta$ on $U$ with zero boundary conditions admits $L^2(U)$-orthonormal eigenfunctions $\{\phi_n:\overline{U}\to\mathbb{R}\}_{n=1}^{\infty}$, corresponding to positive eigenvalues listed in increasing order. $\phi_1$ is called the principal Dirichlet Laplacian eigenfunction, and is everywhere positive on $U$. Moreover, results of Brascamp-Lieb imply that $\phi$ is log-concave, so that $\phi$ achieves its maximum value at some unique $x_U\in U$.
Question: I am looking for a reference to validate or disprove the following claim: The point $x_U$ achieves the inradius $\text{inrad}(U)=\max_{x\in U}\text{dist}(x,\partial U)$ of $U$. (Note that, however, the inradius need not be achieved at a unique point. For example, a long rectangle contains an entire line segment achieving the inradius.)