Cioabă and Wong's spectral spanning-tree packing conjecture

Let $m\geq 1$ be an integer, and let $G$ be a $d$-regular graph with $d\geq 2m+2$. Let $\lambda_2(G)$ be the second-largest adjacency eigenvalue of $G$, and let $\sigma(G)$ denote the maximum number of edge-disjoint spanning trees in $G$. Cioabă and Wong's conjecture. If

$$\lambda_2(G)<d-\frac{2m+1}{d+1},$$

then

$$\sigma(G)\geq m+1.$$

This conjecture gives a factor-of-two improvement over the earlier spectral condition implying the existence of $m+1$ edge-disjoint spanning trees; it was verified for $m\in\{1,2\}$, while the paper constructs examples showing that the conjectured bound is essentially best possible.