Equichordal projection conjecture for convex bodies

Let $K,L\subset\mathbb R^3$ be convex bodies with $L\subset\operatorname{int}K$, and suppose that $L$ is strictly convex. For $u,w\in\mathbb S^2$ with $u\bot w$, let $\pi_u$ denote orthogonal projection in direction $u$. Equichordal projection conjecture. If the chords of $\pi_u(K)$ tangent to $\pi_u(L)$ and parallel to $w$ all have length $\lambda(w)$, then $K$ and $L$ are homothetic and concentric ellipsoids. This extends the preceding theorem, which proves the special case in which all tangent projected chords have a common length. The conjecture remains open in the stated generality.