The centroid-projection symmetry conjecture

Let $K\subset\mathbb R^n$ be a convex body, with $n\geq 3$, and let $q\in\operatorname{int}K$. For each $u\in\mathbb S^{n-1}$, let $\pi_u$ denote orthogonal projection in direction $u$. Centroid-projection conjecture. If $\pi_u(q)$ is the centroid of $\pi_u(K)$ for every $u\in\mathbb S^{n-1}$, then $q$ is the centre of symmetry of $K$. This conjecture is identified as the missing ingredient for extending the projection floating-body result from dimension three to arbitrary dimension; no general proof is supplied.