Continuous spectrum-shrinking maps between finite-dimensional power-associative algebras

For finite-dimensional complex power-associative algebras $\mathcal{A}$ and $\mathcal{B}$, each equipped with its unique Hausdorff vector topology, we characterize continuous spectrum-shrinking maps $ϕ:\mathcal{A}\to\mathcal{B}$, i.e. maps satisfying $\mathrm{sp}_{\mathcal{B}}(ϕ(x)) \subseteq\mathrm