Kippenhahn's conjecture on hermitian matrix algebras

Let $H$ and $K$ be hermitian $2n\times 2n$ matrices, and let

$$f=\det(xH+yK+I)\in\mathbb{R}[x,y].$$

Let $\mathcal{A}$ be the algebra generated by $H$ and $K$. If there exists $g\in\mathbb{C}[x,y]$ such that $f=g^k$, then there is a unitary matrix $U$ such that

$$U^{*}(xH+yK)U$$

is block diagonal, and hence $\mathcal{A}\neq M_n(\mathbb{C})$.

Kippenhahn's conjecture. The algebra generated by $H$ and $K$ cannot be the full algebra $M_{2n}(\mathbb{C})$.

The conjecture concerns the relationship between repeated factors in the determinant polynomial of a linear pencil of hermitian matrices and reducibility of the algebra they generate. The surrounding argument establishes that, in the considered structured setting, every real linear combination $xH+yK$ has eigenvalues of even multiplicity; the supplied text does not state whether the conjecture has been resolved.