  A $3$-dimensional polytope $P$ is $k$-equiprojective when the projection of
$P$ along any line that is not parallel to a facet of $P$ is a polygon with $k$
vertices. In 1968, Geoffrey Shephard asked for a description of all
equiprojective polytopes. It has been shown recently that the number of
combinatorial types of $k$-equiprojective polytopes is at least linear as a
function of $k$. Here, it is shown that there are at least $k^{3k/2+o(k)}$ such
combinatorial types as $k$ goes to infinity. This relies on the
Goodman--Pollack lower bound on the number of order types and on new
constructions of equiprojective polytopes via Minkowski sums.
