The conjecture on extremal normalized distance-Laplacian spectral radius

For integers $n_1,n_3\geq 1$ and $n_2\geq 2$, let $KPK_{n_1,n_2,n_3}$ be formed by taking the vertex sum of a vertex in $K_{n_1}$ with one end of $P_{n_2}$ and the vertex sum of a vertex in $K_{n_3}$ with the other end; it has $n_1+n_2+n_3-2$ vertices. For a connected graph $G$, let $\mathcal D^{\mathcal L}(G)$ be its normalized distance Laplacian matrix and let $\rho$ denote spectral radius. Extremal normalized distance-Laplacian conjecture. The maximum $\mathcal D^{\mathcal L}$ spectral radius among graphs on $n$ vertices tends to $2$ as $n\to\infty$ and is achieved by $KPK_{n_1,n_2,n_3}$ for some $n_1+n_2+n_3=n+2$.

The conjecture addresses the unresolved extremal graph for the normalized distance-Laplacian spectral radius; computations in the survey show that graphs of this form are extremal for $n\leq 10$.