Pólya–Szegő capacity conjecture in the plane

Prove that every compact planar set has Newtonian capacity at most [?] the factor \(2/\pi\) times its logarithmic capacity, with equality for disks, thereby resolving the planar Pólya–Szegő conjecture.

Background

The paper studies the conjecture that, among planar compact sets with prescribed logarithmic capacity, the disk maximizes Newtonian capacity. The conjectured inequality is $$\operatorname{Cap}_1(K)\leq (2/\pi)\operatorname{Cap}_0(K)$$, with equality for disks.

The authors develop a conditional approach through insulated strip energies and show that the conjecture would follow from a sharp transverse Dirichlet-energy inequality for equilibrium potentials in a Neumann strip. The conjecture itself is not proved in the paper.