The negative-energy Hill-boundary radius conjecture

Let $M_E$ be the negative-energy configuration space equipped with the Jacobi–Maupertuis metric $d_E$, let $\partial M_E$ denote its Hill boundary, and let $0$ be the origin configuration. Negative-energy radius conjecture. For $E<0$, the greatest distance from a point of $M_E$ to the Hill boundary is attained at the origin:

$$\sup_{\{q \in M_E\}} d_{E} (\partial M_E, q) = d_E (\partial M_E, 0).$$

This conjecture concerns the sharp radius of the negative-energy configuration space relative to its Hill boundary. The surrounding discussion explains that an earlier proof of a related finite-diameter estimate was incorrect because the proposed bound depended on an unbounded point on the Hill boundary; the conjecture itself is presented without a stated resolution.