Symmetric minimal covering body classification conjecture

Let $K$ be an origin-symmetric convex body in $\mathbb{R}^3$, and suppose that $K$ is a $\mathbb{Z}^3$-minimal covering body. Let $\Delta_i$ denote the standard Kuhn simplices in $\mathbb{R}^3$.

Symmetric minimal covering body classification conjecture. There exists a unimodular transformation $T$ such that

$$T(K)=\operatorname{conv}\{\Delta_i+\mathbf{x}_i\}$$

for some translations $\mathbf{x}_i\in\mathbb{R}^3$.

The conjecture is proposed as a complete classification of three-dimensional centrally symmetric minimal covering bodies, contrasting with the infinitely many types established in higher-dimensional centrally symmetric and three-dimensional asymmetric settings.