The conjecture on the unique minimizer of normalized distance-Laplacian spectral radius

Let $G$ be a connected graph on $n\geq 2$ vertices, let $K_n$ be the complete graph, and let $\rho(\mathcal D^{\mathcal L}(G))$ denote the spectral radius of the normalized distance Laplacian. Unique-minimizer conjecture.

$$\rho\left(\mathcal D^{\mathcal L}(G)\right)=\frac{n}{n-1}$$

if and only if $G$ is the complete graph $K_n$.

The lower bound and the corresponding normalized distance-Laplacian spectrum are known, but uniqueness of the minimizer had not been shown in the survey; the conjecture proposes that only $K_n$ attains equality.