The low-degree Ehrhart coefficient conjecture for lattice-face polytopes

Let $P$ be a $d$-polytope with vertex set $V$, and let $d'\le d-1$. Suppose that for every integer $k$ with $0\le k\le d'$, condition (dfcon) is satisfied. Write $i(P,m)$ for the Ehrhart polynomial of $P$, and let $\pi^{d-d'}(P)$ denote the projection of $P$ to the corresponding $(d')$-dimensional coordinate subspace. Low-degree Ehrhart coefficient conjecture. For every $k$ with $0\le k\le d'$, the coefficient of $m^k$ in $i(P,m)$ is the same as the coefficient of $m^k$ in $i(\pi^{d-d'}(P),m)$. Equivalently, there are coefficients $c_{d'+1},\ldots,c_d$ such that

$$i(P,m)=i(\pi^{d-d'}(P),m)+\sum_{i=d'+1}^{d}c_i m^i.$$

This conjecture predicts that the Ehrhart coefficients through degree $d'$ are determined by the indicated projection whenever the lattice-face condition holds in dimensions $0$ through $d'$. The supplied text gives motivation from examples but no resolution, so the conjecture remains open.