NP-hardness conjecture for limited-capacity covering and packing

Let $C$ be a convex body and let $k\geq 3$ be an integer. The decision problem $C$-$k$-COVER asks whether a given point set can be covered by homothets of $C$ with each homothet containing at most $k$ points; similarly, $C$-$k$-PACK asks whether a prescribed number of homothets of $C$ can be selected subject to each homothet containing at most $k$ points.

NP-hardness conjecture. $C$-$k$-COVER is NP-hard for every convex body $C$ and every integer $k\geq 3$. Similarly, $C$-$k$-PACK is NP-hard for every convex body $C$ and every integer $k\geq 2$.

The conjecture extends the paper's NP-hardness results beyond the particular cases handled by its reduction to $3$-SAT and by earlier work. Its status is not resolved in the supplied source context.