Internal zonotopal space interpolation conjecture for box splines

Let $X$ be a unimodular list of vectors, let $Z(X)$ be its zonotope, and let ${\cal Z}_-(X)$ be the set of integer points in the interior of $Z(X)$. Let $M_X$ be the box spline associated with $X$, and let ${\cal I}_-(X)$ be the internal ideal. For a polynomial $p$, write $p(D)$ for the corresponding constant-coefficient differential operator. Internal interpolation conjecture. For every function $f:{\cal Z}_-(X)\to\mathbb{R}$, there exists a unique polynomial $p\in\ker {\cal I}_-(X)$ such that $p(D)M_X$ agrees with $f$ on ${\cal Z}_-(X)$. This would give an interpolation theorem for the values of box-spline differential operators at the interior lattice points of the zonotope; the supplied text does not state whether it has been proved or remains open.