Ehrhart's volume conjecture for lattice-free centered convex bodies

Let $K \subset \mathbb{R}^d$ be a convex body with centroid $\boldsymbol{0}$ and no nonzero lattice point in its interior:

$$\operatorname{int}(K)\cap\mathbb{Z}^d=\\{\boldsymbol{0}\\}.$$

Let $S_d=(d+1)\operatorname{conv}\\{\boldsymbol{0},\boldsymbol{e}_1,\ldots,\boldsymbol{e}_d\\}-\boldsymbol{1}$.

Ehrhart's volume conjecture. One should have

$$\operatorname{vol}(K)\leq\operatorname{vol}(S_d)=\frac{(d+1)^d}{d!}.$$

This is the volume analogue related to the preceding lattice-point conjecture and is attributed in the source to Ehrhart. The simplex $S_d$ gives the proposed extremal example; the source does not indicate a resolution.