Common eigenvectors and quasicommutativity of sets of simultaneously triangularizable matrices

A set of simultaneously triangularizable square matrices over an arbitrary field is considered. If the matrices are also quasicommutative, then they have a common eigenvector for every distinct set of corresponding eigenvalues. Conversely, if the set of matrices has this common eigenvector property hereditarily (i.e., for every set of corresponding blocks in every simultaneous block triangularization), then the matrices are quasicommutative.