Ehrhart-function criterion for arbitrary equidecomposability

Let $S$ and $S'$ be bounded subsets of $\mathbb{R}^2$ with $\Delta$-complex decompositions $\mathcal{T}$ and $\mathcal{T}'$, respectively. Let $\operatorname{ehr}_S(t)$ denote the Ehrhart function of $S$. An equidecomposability relation is a map

$$\mathcal{F}:(S,\mathcal{T})\to(S',\mathcal{T}').$$

Ehrhart-function conjecture. There exists an equidecomposability relation $\mathcal{F}:(S,\mathcal{T})\to(S',\mathcal{T}')$ if and only if

$$\operatorname{ehr}_S(t)=\operatorname{ehr}_{S'}(t).$$

The conjecture proposes that the Ehrhart function completely characterizes equidecomposability when arbitrary decompositions, including potentially infinite or otherwise unrestricted ones, are allowed. The source gives no resolution.