Conformal maximality conjecture for free boundary harmonic annuli

For any $T\geqslant T_1$, let $\mathbb{A}_T=[0,T]\times\mathbb{S}^1$ and let $\mathcal{C}_T$ be the conformal class containing the flat metric on $[0,T]\times\mathbb{S}^1$. Let $\hat\Psi_T\colon\mathbb{A}_T\to\mathbb{B}^3$ be the free boundary harmonic map obtained in the theorem, and let $E(\hat\Psi_T)$ denote its energy. Conformal maximality conjecture. Then

$$\sup_{g\in\mathcal{C}_T}\bar\sigma_1(\mathbb{A}_T,g)=2E(\hat\Psi_T).$$

In particular, for these conformal classes, $\bar\sigma_1$-conformally maximal pairs can be chosen to be rotationally symmetric. This is the precise formulation of the conjectured maximality of the constructed annular metrics, and it remains open in the supplied text.