The asymptotic constant conjecture for the operator determinant inequality

Let $k_n$ be the smallest constant such that, for every $n$-dimensional normed space $X$ and every invertible operator $T\in\mathcal L(X)$,

$$|\det(T)|\,\|T^{-1}\|\leq k_n\|T\|^{n-1}.$$

The known bounds are

$$\sqrt n(1-o(1))\leq k_n\leq\sqrt{en}.$$

Asymptotic constant conjecture. There exists a constant $1\leq c\leq\sqrt e$ such that

$$k_n\sim c\sqrt n.$$

This conjecture asks whether the normalized constants $k_n/\sqrt n$ converge. The existing bounds determine only that their eventual size lies between $1$ and $\sqrt e$.