Hermitian pencils with a cubic minimal polynomial

Let A be an n × n complex matrix and write A = H + iK, where H and K are Hermitian matrices. We show that if the minimal polynomial of the pencil xH + yK has degree 3, then there is a unitary matrix U such that U^-1AU is block diagonal with blocks of size 3 × 3 or smaller. This is a special case of a conjecture made by Kippenhahn in 1951.