The simplex conjecture for covering shadows

Let $K,L\in\mathcal{K}_n$ be convex bodies, and for a unit vector $u\in\mathbb{R}^n$ let $K_u$ and $L_u$ denote their projections onto the hyperplane orthogonal to $u$. Suppose that, for every unit vector $u\in\mathbb{R}^n$, the projection $L_u$ contains a translate of $K_u$. The simplex conjecture. Then

$$V_n(K)\leq\frac{n}{n-1}V_n(L).$$

The bound is known in dimension $2$, where the best possible universal constant is $3/2$, but remains open for dimensions $n\geq3$. If true, it would imply that the universal volume-ratio constant in all dimensions $n\geq2$ is $3/2$; the conjectured bound is not expected to be sharp separately in dimensions $n\geq3$.