Reinhart's normalized distance Laplacian spectral-radius characterization

Let $G$ be a graph on $n$ vertices, and let $\partial_n^\mathcal{L}$ denote the largest eigenvalue of its normalized distance Laplacian matrix $\mathcal{D}^\mathcal{L}(G)$. Reinhart's conjecture.

$$\partial_n^\mathcal{L}=\frac{n}{n-1}$$

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

Reinhart proposed this characterization after proving the bound $\partial_n^\mathcal{L}\geq n/(n-1)$ for graphs on at least two vertices. The supplied source does not indicate whether the conjecture has been resolved.