Let $K \subset \mathbb{R}^3$ be a convex body. Assume that all orthogonal projections of $K$ onto $2$-dimensional subspaces have a center of symmetry. Does it follow that $K$ must also have a center of symmetry?

The question that came up in some work I'm doing right now is just a bit different:

Let $K \subset \mathbb{R}^{2n}$ be a convex body. Assume all orthogonal projections of $K$ onto Lagrangian subspaces have center of symmetry. Does it follow that $K$ must also have a center of symmetry?

Note. The center of symmetry of the shadows may depend on the subspace containing it.