The log-concave spectrum conjecture for Frobenius maximal parabolic type-A seaweeds

Let $\mathfrak{g}$ be a Frobenius maximal parabolic type-A seaweed, meaning a Frobenius seaweed subalgebra of $\mathfrak{sl}(n)$ obtained from a maximal parabolic construction. Its spectrum is the multiset of eigenvalues of the adjoint action of a principal element, with multiplicities. Log-concave spectrum conjecture. The spectrum of $\mathfrak{g}$ has the log-concave spectrum property. The paper observes that log-concavity fails for Frobenius type-A seaweeds in general, but computations for maximal parabolic families suggest the stated restriction may force it; the conjecture remains open.