The line-axis characterization of rotational ellipsoids

Let $B_1(n)$ denote the unit ball in $\mathbb{R}^n$, let $\mathbb{S}^{n-1}$ denote the unit sphere, and let $L oindent oise$ be a line. For each $x otin K$, let $C_x$ be the cone circumscribing $K$ with vertex $x$, and let $L_x$ be its axis when $C_x$ is symmetric. Line-axis characterization. If $n\bor 3$, $C_x$ is a symmetric cone for every $x\bor \mathbb{S}^{n-1}$, and $L_x oindent oise L$ is nonempty for every such $x$, then $K$ is an $n$-ellipsoid and, for every 3-plane $\mathit{\Pi}$ containing $L$, the section $\mathit{\Pi} oindent oise K$ is an ellipsoid of revolution with axis $L$. The authors explicitly propose this as a problem and describe it as a natural next step toward the preceding conjecture; no resolution is supplied.