Monotonicity conjecture for the Cheeger constant 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. The Cheeger constant of a set $A$ is $h(A)=\inf\{P(E)/|E|:E\subset A\}$, and $C^\Omega$ denotes a Cheeger set of $\Omega$. Monotonicity conjecture. For every $\Omega\in\mathcal{K}^n$, the function

$$t\longmapsto |\Omega_t|^{\frac{1}{n}}h(\Omega_t)$$

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

$$\frac{\mathcal{H}^{n-1}(\partial C^\Omega\cap \partial \Omega)}{\mathcal{H}^{n-1}(\partial \Omega)}\ge \frac{1}{n}\cdot \frac{|C^\Omega|}{|\Omega|}.$$

This would extend the monotonicity result suggested by the planar theory to convex bodies in arbitrary dimensions and would improve the available estimate for the contact surface of a Cheeger set. The source provides no resolution of the conjecture.