Monotonicity conjecture for the normalized Cramér-transform exponential functional

For each $n\in\mathbb{N}$, let $X_n$ be the rotationally invariant random vector denoted by $\mathcal{E}$ in $\mathbb{R}^n$, and let $\Lambda_{X_n}^\ast$ be its Cramér transform. Normalized exponential-functional monotonicity conjecture. The function

$$n\longmapsto\left(\mathbb{E}e^{-\Lambda_{X_n}^\ast(X_n)}\right)^{1/n}$$

is non-increasing. The conjecture is motivated by numerical evidence and would identify the dimension dependence of the sharp constant in the rotationally invariant one-shot separability problem. The supplied text does not report a proof or counterexample.