Gehring–Martin–Palka conjecture on the conformal capacity of linked curves

Let C_0,C_1\binom{\mathbb S^3} be linked curves, and let $H_0,H_1$ be the components of the standard Hopf link. The relative conformal capacity is

$$\operatorname{Cap}_3(C_0,C_1)=\inf_u\int_{\mathbb S^3}|\mathrm{d}u|^3\,\operatorname{dVol},$$

where the infimum is over absolutely continuous functions $u\colon\mathbb S^3\to[0,1]$ with $u=0$ on $C_0$ and $u=1$ on $C_1$.

Gehring–Martin–Palka conjecture. Every pair of linked curves satisfies

$$\operatorname{Cap}_3(C_0,C_1)\geq\operatorname{Cap}_3(H_0,H_1)=\frac{16\pi^3}{\Gamma(1/4)^4},$$

where $\Gamma(t)$ is the Gamma function.

The conjecture proposes that the standard Hopf link minimizes conformal capacity among linked curves. Its status is not resolved in the supplied source.