Saroglou's upper-bound conjecture for the second projection body of zonoids

Let $K$ be a zonoid in $\mathbb{R}^d$, let $V(K)$ be its volume, and let $\Pi^2K:=\Pi(\Pi K)$ be its second projection body. Saroglou's upper-bound conjecture.

$$\Pi^2K\subseteq 2^dV(K)^{d-2}K,$$

with equality if and only if $K$ is a Weil-convex body. In dimension $3$, the source proves the corresponding bound for zonoids, with equality exactly for centrally symmetric cylinders; the all-dimensional statement remains conjectural in the source.