Petty's polar projection problem

Let $K\subset\mathbb{R}^n$ be an $o$-symmetric convex body, and let $\Pi K$ be its projection body, defined by

$$h_{\Pi K}(u)=\operatorname{vol}_{n-1}(K\mid u^\perp),\qquad u\in\mathbb{S}^{n-1}.$$

Its polar projection body is $\Pi^\circ K=(\Pi K)^\circ$. Petty's conjecture. If $n\geq 3$ and $\Pi^\circ K=\lambda K$ for some $\lambda\in\mathbb{R}$, then $K$ is an ellipsoid. This polar projection problem is a classical question in convex geometry; the stated version is known for convex polytopes, where Martini proved that the only such polytopes are simplices, but the general problem remains open.