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In previous work, Darvas-George-Smith obtained inequalities between the largescale asymptotic of the $J$ functional with respect to the $d_1$ metric on thespace of toric K\"ahler metrics/rays. In this work we prove sharpness of theseinequalities on all toric K\"ahler manifolds, and study the extremizingpotentials/rays. On general K\"ahler manifolds we show that existence of radialextremizers is equivalent with the existence of plurisupported currents, asintroduced and studied by McCleerey.

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