$\pi^{\prime}$ is a piecewise-linear map, thus $\pi$ pulls back any piecewise-linear function on $\Sk(X)$ to a model function on $X^{\an}$;

$\pi$ is an affinoid torus fibration away from a graph $\Gamma\subset\Sk(X)$; the vertices of $\Gamma$ are the barycenters of the 1 and 2-dimensional cells of $\Sk(X)$, and the edges join the barycenter of a 2-dimensional cell with the barycenters of its 1-dimensional faces;

in a neighbourhood of a vertex $v_{i}\in\Sk(X)$, the affine structure induced by $\pi$ is determined by the toric geometry of $D_{i}$: there is a natural $\mathbb{Z}$-linear embedding of $\Star(v_{D_{i}})$ inside the fan of $D_{i}$, preserving the polytopal decomposition and sending $v_{D_{i}}$ to the origin;

$\pi$ induces an integral affine structure on $\Sk(X)\setminus\Gamma$, which is isomorphic to the ones constructed in [Gro01] and [Rua01], and in [Li19] for Fermat families.