The maximum edge derivative conjecture for normalized Laplacian eigenvalues

Let $G$ be a graph, let $x,y$ be vertices, and let $\lambda$ be an eigenvalue of the normalized Laplacian whose edge derivative is being considered. The edge derivative with respect to the pair $\{x,y\}$ is denoted by $\frac{d\lambda}{d\{x,y\}}$. Maximum edge derivative conjecture. Over all graphs, the maximum edge derivative for any pair of vertices $x,y$ is $1$; i.e.

$$\max_{G,\{x,y\}} \frac{d\lambda}{d\{x,y\}}=1.$$

The bound is tight for the graph described in the surrounding discussion, and the observed extremal structure involves connected twins. Whether the bound holds for all graphs remains open.