Equichordal projection conjecture for a point

Let $K\subset\mathbb R^3$ be a convex body and let $p$ be a point in its interior. For $u\in\mathbb S^2$, let $\pi_u$ denote orthogonal projection in direction $u$. Point equichordal projection conjecture. If, for every $u\in\mathbb S^2$, the point $\pi_u(p)$ is an equichordal point of $\pi_u(K)$, then $K$ is a ball. This is the point-inner-body case suggested after the projected-body result, and the paper indicates that a subsequent theorem provides partial progress toward a proof. The conjecture remains open.