  We investigate the set of quantum states that can be shown to be
$k$-incoherent based only on their eigenvalues (equivalently, we explore which
Hermitian matrices can be shown to have small factor width based only on their
eigenvalues). In analogy with the absolute separability problem in quantum
resource theory, we call these states "absolutely $k$-incoherent", and we
derive several necessary and sufficient conditions for membership in this set.
We obtain many of our results by making use of recent results concerning
hyperbolicity cones associated with elementary symmetric polynomials.
