Makeev's universal covering conjecture for convex bodies

Let $K$ be a convex body in $\mathbb{R}^d$ of diameter at most $1$. For a non-degenerate simplex

$$\Delta=\operatorname{conv}\{a_0,a_1,\ldots,a_d\}$$

of diameter at most $1$ satisfying $a_0+\cdots+a_d=0$, define its $\Delta$-zonotope by

$$R_\Delta=[0,a_0]+[0,a_1]+\cdots+[0,a_d].$$

Makeev's conjecture. Every convex body $K$ in $\mathbb{R}^d$ of diameter $\leq 1$ can be covered by a translate of $R_\Delta$ for some non-degenerate simplex $\Delta$ of diameter $\leq 1$. This is a universal-covering problem for convex bodies and generalizes the planar covering result by a suitable $\Delta$-zonotope. The source presents the assertion as a problem of Makeev; no resolution is supplied in the given text.