On Packing with Axi-symmetric Bodies

Question:

Consider any 3D body with an axis of rotational symmetry (e.g. cone, cylinder...) and packing the 3d space efficiently with infinitely many copies of this body...
Can one say that "the densest packing with any such body is necessarily such that all units are aligned along or opposite to the same direction"?

Proving such a claim will greatly limit the possibilities that need to be considered to find the densest packing.
Shall update this page when more info comes my way

The 2D analog of the claim above would involve bodies with a reflection symmetry.

Note: In an earlier post here (http://nandacumar.blogspot.com/2018/04/semicircle-packing.html), one had wondered: Is it possible to generalize Fejes Toth's result to say (say!) "a lattice pattern is the best way to pack convex sets with an axis of reflection symmetry"?

Update(27th July 2021): The above questions have been answered here: https://mathoverflow.net/questions/397624/on-packing-axisymmetric-bodies-in-3d