Existence of mono-monostatic convex bodies in smooth strictly convex normed spaces

Let $\mathbb{M}^3$ be a $3$-dimensional normed space whose unit ball has a $C^2$-class differentiable boundary and strictly positive Gaussian curvature. Existence conjecture. There is a mono-monostatic convex body in $\mathbb{M}^3$. The conjecture asks whether the rotational-symmetry hypothesis in the three-dimensional approximation theorem is merely technical. It is presented as an extension of that theorem to all normed spaces satisfying the stated smoothness and curvature conditions.