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

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