The affine structure we obtain in Theorem A coincides also with the one defined in [Gro05, §1]. Indeed, it will follow from the construction and Corollary 2.6.1 that the affine structure induced by $\pi$ yields the fan structure (in the sense of [Gro05]) coming from $D_{i}$ at each vertex $v_{D_{i}}$, and that those are glued (after removing $\Gamma$) along the maximal cells viewed as standard simplices; this is the very definition of the singular affine structure in [Gro05].
In particular, the results of Section 4.7 follow from the more general [Gro05, Proposition 2.13], [HZ02, §2.3], but we include our full computations to highlight the use of Proposition 3.2.2, which holds even outside the context of toric degenerations.