Polarization monotonicity conjecture for the transfinite diameter

Let $E$subset ${\mathbb R}^2$ be compact and let $H$ be a polarizer. The polarization monotonicity conjecture.

$$T_{diam}(P_H(E))\leq T_{diam}(E).$$

Moreover, equality holds only if $P_H(E)\cong E$ or $P_H(E)\cong \sigma_H(E)$. This extends the corresponding monotonicity known for Schwarz and Steiner symmetrizations; the conjectured polarization inequality remains open.