Saroglou's lower-order projection-body minimizer conjecture

Let $K$ be a convex body in $\mathbb{R}^d$. For $i=1,\dots,d-1$, define the order-$i$ projection body $\Pi_iK$ using the mixed projection-body construction, and set

$$P_i(K):=\frac{V(\Pi_iK)}{V(K)^i},\qquad Q_i(K):=\frac{V\big(\Pi_i(\Pi K)\big)}{V(K)^{i(d-1)}}.$$

Saroglou's lower-order projection-body conjecture. For $i=1,\dots,d-2$, the functionals $P_i(\cdot)$ and $Q_i(\cdot)$ are minimized precisely for Euclidean balls. The source notes that the $i=d-1$ case of $P_i$ is Petty's conjecture and that this statement is open; it also identifies the $Q_1$ case as a conjecture of Maresch and Schuster.