  Let $X$ be a compact K\"ahler manifold. We study plurisupported currents on
$X$, i.e. closed, positive $(1,1)$-currents which are supported on a pluripolar
set. In particular, we are able present a technical generalization of
Witt-Nystr\"om's proof of the BDPP conjecture on projective manifolds, showing
that this conjecture holds on $X$ admitting at least one plurisupported current
$T$ such that $[T]$ is K\"ahler.
  One of the steps in our proof is to show an upper-bound for the pluripolar
mass of certain envelopes of quasi-psh functions when the cohomology class is
shifted, a result of independent interest. Using this, we are able to
generalize an inequality of McKinnon and Roth to arbitrary pseudoeffective
classes on compact K\"ahler manifolds.
