Monotonicity conjecture for the first Dirichlet eigenvalue of parallel bodies

Let $\Omega\in\mathcal{K}^n$ be a convex body, let $r(\Omega)$ denote its inradius, and write $\Omega_t=\Omega\oplus tB_1$ for its parallel body. Let $\lambda_1(\Omega)$ be the first Dirichlet eigenvalue of the Laplacian on $\Omega$, and let $u_\Omega$ be its first normalized eigenfunction. Monotonicity conjecture. For every $\Omega\in\mathcal{K}^n$, the function

$$t\longmapsto |\Omega_t|^{\frac{2}{n}}\lambda_1(\Omega_t)$$

is monotonically decreasing on $(-r(\Omega),+\infty)$. As a consequence,

$$\frac{1}{P(\Omega)}\int_{\partial\Omega}|\nabla u_\Omega|^2d\mathcal{H}^{n-1}\ge \frac{2}{n}\cdot\frac{1}{|\Omega|}\int_{\Omega}|\nabla u_\Omega|^2d\mathcal{H}^n,$$

with equality if and only if $\Omega$ is a tangential body. Numerical simulations suggest that this analogue of the Cheeger-constant monotonicity theorem may hold, but the source provides no proof or resolution.