The conjecture on Steklov functional maximizers for critical ellipses

Let $\mathbb{D}$ be the disk, let $h_t^+$ denote the corresponding Steklov data for the critical ellipse, let $E_t=\{x^2+t y^2=1\}$, and let $co(E_t)=\{x^2+t y^2\leq 1\}$. Equip $co(E_t)$ with the Euclidean metric $\xi$ and the boundary weight $\beta(x,y)=(x^2+t y^2)^{-1}$. Ellipse conjecture. For $1\leq t\leq 3$,

$$I^S(\mathbb{D},h_t^+)=\frac{\sqrt{t}}{2\pi}.$$

Moreover, this value is uniquely attained by the critical ellipse $co(E_t)$ with the specified metric and boundary weight. This would improve the description of $I^S(\mathbb{D},h_t^+)$ and its minimizers; the source presents it as an open conjecture related to the existence of non-planar free boundary minimal disks in ellipsoids.