The spectral conjecture for finite-dimensional real *-algebras

Let $(\mathcal A,*)$ be an associative, unital, finite-dimensional $*$-algebra over $\mathbb R$. The spectral monoid $\overline{\operatorname{Her}}(\mathcal A,*)$ is the monoid of Hermitian matrices over $(\mathcal A,*)$ modulo unitary similarity, with direct sum as its operation. The spectral conjecture. For every such $(\mathcal A,*)$, the spectral monoid $\overline{\operatorname{Her}}(\mathcal A,*)$ is a subfree abelian monoid. This would provide a qualitative, potentially non-constructive generalization of several matrix decomposition and spectral-theorem phenomena to arbitrary finite-dimensional real $*$-algebras; the source gives no proof or resolution of the conjecture.