A refinement of the proof shows that the equality $\rho_{A}=\rho_{\mathscr{P}}$ remains valid if we only assume that $A$ has semi-abelian reduction; then the non-archimedean uniformization of $A$ takes the form $\pi:E^{\mathrm{an}}\to A^{\mathrm{an}}$, where $E$ is an extension of an abelian $K$-variety $B$ with good reduction by a split $K$-torus $T$. The dimension of $T$ is precisely the toric rank of $\mathscr{A}^{o}_{k}$, the identity component of the special fiber of the Néron model of $A$. The Künnemann-Mumford construction produces a relatively complete model $\widetilde{\mathscr{P}}$ of $E$ that is a Zariski-locally trivial fibration in torus embeddings over the Néron model of $B$. Since we do not need this generalization in this paper, we omit the details.