Uniqueness of the minimum normalized distance Laplacian spectral radius

Let $G$ be a graph on $n\geq 2$ vertices. Write $\partial^{\mathcal L}_n$ for the largest eigenvalue of its normalized distance Laplacian and $\rho_{\mathcal D^{\mathcal L}}$ for the corresponding spectral radius. Minimum spectral-radius conjecture. For a graph on $n$ vertices,

$$\rho_{\mathcal D^{\mathcal L}}=\partial^{\mathcal L}_n=\frac{n}{n-1}$$

if and only if $G$ is the complete graph $K_n$; consequently, $K_n$ is the only graph achieving the minimum spectral radius with respect to $\mathcal D^{\mathcal L}$. The lower bound is proved in the paper, while uniqueness is supported by computation for $n\leq 20$ and would follow from showing that $K_n$ has no normalized distance Laplacian cospectral mates; the general claim remains open.