Degree and transmission values in quasi-$(\lambda,n)$-distance-balanced graphs

Let $G$ be a quasi-$\lambda$-distance-balanced graph. For a vertex $u$ of $G$, write $\deg(u)$ for its degree and $D(u)=\sum_{v\in V(G)}d(u,v)$ for its total distance (transmission). Degree–transmission conjecture.

$$\left|\{\deg(u):u\in V(G)\}\right|=\left|\{D(u):u\in V(G)\}\right|=2.$$

The conjecture is motivated by the bipartite structure of quasi-$\lambda$-distance-balanced graphs and the observed restrictions on the possible sizes of the sets $W_{uv}^{G}$. Its status is not resolved in the supplied source.