Freedman–He conjecture on the conformal capacity of solid tori

For a solid torus $T\subset\mathbb S^3$, let $T^*=\overline{\mathbb S^3\setminus T}$ be its dual torus, and define

$$\operatorname{Cap}_3(T)=\inf_u\int_T|\mathrm{d}u|^3\,\operatorname{dVol},$$

where the infimum is over absolutely continuous functions $u\colon T\to\mathbb R\setminus\mathbb Z$ of degree one. Let

$$\mathbb T_{\mathrm{sol}}=\{(z_1,z_2)\in\mathbb C^2:|z_1|^2\leq|z_2|^2=1/2\}$$

be the solid Clifford torus.

Freedman–He conjecture. For every solid torus $T\subset\mathbb S^3$,

$$\min\{\operatorname{Cap}_3(T),\operatorname{Cap}_3(T^*)\}\leq\operatorname{Cap}_3(\mathbb T_{\mathrm{sol}})=\frac{\sqrt2-1}{2\pi}.$$

The conjecture asserts that the solid Clifford torus gives the optimal configuration for this dual solid-torus capacity problem. Its status is not resolved in the supplied source.