The face-number conjecture for perfect prismatoids

Let $P$ be a $d$-dimensional perfect prismatoid. A $d$-dimensional lattice Delaunay polytope is a convex polytope whose vertices belong to a lattice and which is the convex hull of the lattice points lying on an empty sphere. For each $k$ with $0\leq k\leq d$, let $f_k(P)$ denote the number of $k$-dimensional faces of $P$. The conjecture concerns the existence of a lattice Delaunay polytope $Q$ of dimension $d$ satisfying the following inequalities.

Face-number conjecture. For every $d$-dimensional perfect prismatoid $P$, there exists a $d$-dimensional lattice Delaunay polytope $Q$ such that

$$f_k(P)\geq f_k(Q)$$

for every $k$ with $0\leq k\leq d$.

This is a proposed strengthening of the comparison between perfect prismatoids and lattice Delaunay polytopes in terms of all face numbers. The supplied source context does not establish whether the conjecture is open or resolved.