The conjecture on non-planar free boundary minimal disks in ellipsoids

For $q\geq 1$, consider the ellipsoid

$$\{x^2+q y^2+q z^2\leq 1\}\subset\mathbb{R}^3.$$

Ellipsoid existence conjecture. There is a non-planar free boundary minimal immersed disk into this ellipsoid if and only if $q>3$. This is presented as a natural conjecture arising from the description of critical ellipses; its resolution is not given in the source.