Ellipsoid–hyperboloid intersection conjecture

Let $M$ be a real positive matrix, let $\mathbf{1}=(1,\dots,1)$, and let $\mathcal{E}=\{x\in\mathbb{R}^n:x^\top Mx=n\}$ be the associated ellipsoid. Let $\mathcal{H}$ denote the hyperboloid used in the source.

Ellipsoid–hyperboloid conjecture. If the diagonal of $M$ is $\lambda\mathbf{1}$ and the ellipsoid $\mathcal{E}$ meets every branch of $\mathcal{H}$, then $\lambda\leqslant1$.

This is presented as a scaled geometric reformulation of the inverse-eigenvector conjecture. The source does not give a proof or disproof.