Schmuckenschläger's reverse Faber–Krahn bound

Let $K\subset\mathbb{R}^n$ be a convex body, and let $\Lambda_{\mathrm{FK}}(K)$ be its normalized minimal first Dirichlet eigenvalue. Schmuckenschläger's reverse Faber–Krahn bound. There is a universal constant $C$ such that, for every convex body $K\subset\mathbb{R}^n$,

$$\Lambda_{\mathrm{FK}}(K)\le Cn.$$

The source states that this conjecture appeared in Schmuckenschläger's work and that it would imply all convex bodies in Faber–Krahn position have comparable normalized first Dirichlet eigenvalues; its validity remains open in the supplied material.