Brouwer's toughness conjecture for regular graphs

Let $G$ be a connected $d$-regular graph, let $t(G)$ denote its toughness, and let $\lambda$ be the second largest absolute eigenvalue of the adjacency matrix of $G$. Brouwer's toughness conjecture. For any connected $d$-regular graph $G$,

$$t(G)\ge \frac{d}{\lambda}-1.$$

Brouwer's conjecture strengthens his eigenvalue bound $t(G)>\frac{d}{\lambda}-2$; partial results are known, but the source states that no substantial progress had been made for more than two decades before the paper's proof.