van den Berg's localization conjecture for first eigenfunctions of convex domains

Let $\Omega\subset\mathbb{R}^n$ be a bounded convex domain, and let $u$ be a first Dirichlet eigenfunction of $\Omega$. Write $\operatorname{diam}(\Omega)$ for its diameter and $\operatorname{inrad}(\Omega)$ for its inner radius. van den Berg's conjecture. There exists a constant $c_n>0$, depending only on the dimension $n$, such that

$$\lVert u\rVert_{L^{2}(\Omega)} \geq c_n \left(\operatorname{diam}(\Omega)/\operatorname{inrad}(\Omega)\right)^{1/6}\operatorname{inrad}(\Omega)^{n/2}\lVert u\rVert_{L^{\infty}(\Omega)}.$$

This conjecture quantifies the expected spreading of the first eigenfunction in a convex domain whose diameter is large relative to its inner radius. The two-dimensional case has been established, while the conjecture in higher dimensions remains open.