Edge-disjoint spanning trees and eigenvalues

Let τ(G) and λ2(G) be the maximum number of edge-disjoint spanning trees and the second largest eigenvalue of a graph G, respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and τ(G), Cioabă and Wong conjectured that for any integers k⩾2, d⩾2k and a d-regular graph G, if λ2(G)