The equality $\Delta(A)=\mathrm{Sk}(A)$ is proven in [HN17, 4.3.2]. Let $\mathscr{P}$ be a Künnemann-Mumford model for $A$ over $R$. Then, by definition, $\mathscr{P}$ is an snc-model, and thus certainly good and dlt. It is shown in [HN17, 5.1.7] that $\mathscr{P}$ is minimal.

Let $\widetilde{\mathscr{P}}$ be a regular relatively complete model of $T$ as in [Kü98, 2.11] such that the formal $t$-adic completion of $\mathscr{P}$ arises as a quotient of the formal $t$-adic completion of $\widetilde{\mathscr{P}}$ under an action of the period lattice. Then, by construction, $\widetilde{\mathscr{P}}$ is a torus embedding of $T$ over $R$, and we have a commutative diagram

Thus in order to prove that $\rho_{\mathscr{P}}=\rho_{A}$, it suffices to observe that $\rho_{\widetilde{\mathscr{P}}}=\rho_{T}$ by Example 3.5. ∎