  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 K\"ahler metrics/rays. In this work we prove sharpness of these
inequalities on all toric K\"ahler manifolds, and study the extremizing
potentials/rays. On general K\"ahler manifolds we show that existence of radial
extremizers is equivalent with the existence of plurisupported currents, as
introduced and studied by McCleerey.
