Ehrhart's multivariate quasi-polynomial conjecture

Let $A(n_1,\dots,n_r)$ be an $m\times k$ matrix and $b(n_1,\dots,n_r)$ be a column vector of length $m$, with all entries linear functions of $n_1,\dots,n_r$ having integer coefficients. Suppose that, for every $n_1,\dots,n_r$, the number of nonnegative integer solutions $x$ to

$$A(n_1,\dots,n_r)x=b(n_1,\dots,n_r)$$

is finite. A function $f$ is the number of such solutions. Ehrhart's conjecture. There exist finitely many polyhedral regions $R_1,\dots,R_N$ covering $\mathbf{R}^r_{\ge 0}$ such that $f$ is a multivariate quasi-polynomial when restricted to each $R_i$. Here, a function on a subset of $\mathbf{Z}^r$ is a multivariate quasi-polynomial if some finite-index sublattice of $\mathbf{Z}^r$ makes it polynomial on each intersected coset. The conjecture is a multivariable analogue of Ehrhart-type quasi-polynomiality results and predicts piecewise multivariate quasi-polynomial behavior for parametric systems of linear Diophantine equations; the supplied text does not establish its resolution.