Brandt's adjacency-spectrum conjecture for regular triangle-free graphs

Let $G$ be a regular triangle-free graph on $n$ vertices. Let $\lambda_n(G)\leq\cdots\leq\lambda_1(G)$ be the eigenvalues of the adjacency matrix of $G$. Brandt's conjecture.

$$\lambda_1(G)+\lambda_n(G)\leq \frac{4}{25}\cdot n.$$

The source presents this as a conjecture connected to the problem of making triangle-free graphs bipartite and gives Brandt's spectral inequality, but supplies no resolution of the conjecture.