Extremizers of the $J$ functional with respect to the $d_1$ metric

In previous work, Darvas-George-Smith obtained inequalities between the large scale asymptotic of the $J$ functional with respect to the $d_1$ metric on the space of toric Kähler metrics/rays. In this work we prove sharpness of these inequalities on all toric Kähler manifolds, and study the extremizing potentials/rays. On general Kähler manifolds we show that existence of radial extremizers is equivalent with the existence of plurisupported currents, as introduced and studied by McCleerey.