Spectral gap of the normalized distance Laplacian

The smallest positive eigenvalue $\partial_2$ of the normalized distance Laplacian matrix $\mathcal{D}^{\mathcal{L}}$ of a connected graph is called its \emph{spectral gap} and is intimately related to the Cheeger constant of $\mathcal{D}^{\mathcal{L}}$. Byrne, Johnston, Schildkraut and Tait (2025) conjectured that \[ \partial_2 \ge \frac{2}{3}\] for all connected graphs. We prove the following stronger result: for any connected graph $G$ of order at least 2, \[\partial_2 \ge \frac{2}{3} + \frac{4}{3\,t_{\max}},\] where $t_{\max}$ denotes the maximum transmission in $G$. Moreover, equality holds if and only if $G\cong K_{m,m}$ for some $m\ge 1$.