Disproof of a conjecture on the main spectrum of generalized Bethe trees

An eigenvalue of the adjacency matrix a graph is said to be main if all-ones vector not orthogonal its associated eigenspace. A generalized Bethe tree with $k$ levels rooted in which vertices at same level have degree. Fran\c{c}a and Brondani [On spectrum trees, Linear Algebra Appl., 628 (2021) 56-71] recently conjectured that any has exactly eigenvalues whenever even. We disprove conjecture by constructing family counterexamples for even integers $k\ge 6$.