Finite similar shapes conjecture for MICP representable sets

Let $S\subseteq\mathbb{R}^n$ be MICP representable, so that

$$S=\bigcup_{\bm{z}\in C\cap\mathbb{Z}^d}A_{\bm{z}},$$

where $C\subseteq\mathbb{R}^d$ and each $A_{\bm{z}}\subseteq\mathbb{R}^d$ is a projection of a closed convex set. A finite family of similar shapes should suffice: there should exist a finite set $C_0\subseteq C\cap\mathbb{Z}^d$ such that, for every $\bm{z}\in C\cap\mathbb{Z}^d$, some $\bm{z}_0\in C_0$ satisfies that $A_{\bm{z}}$ is homothetic to $A_{\bm{z}_0}$, meaning that it is a translation and scaling of $A_{\bm{z}_0}$. This conjecture proposes a finite-shape refinement of the equal-shape property for MICP representations: although the projected sets need not all be translates of finitely many convex sets when their volumes vary infinitely, they should differ from finitely many model sets only by translations and scalings.