  In \cite{GL21a} we have developed a new approach to $L^{\infty}$-a priori
estimates for degenerate complex Monge-Amp\`ere equations, when the reference
form is closed. This simplifying assumption was used to ensure the constancy of
the volumes of Monge-Amp\`ere measures.
  We study here the way these volumes stay away from zero and infinity when the
reference form is no longer closed. We establish a transcendental version of
the Grauert-Riemenschneider conjecture, partially answering conjectures of
Demailly-P\u{a}un \cite{DP04} and Boucksom-Demailly-P\u{a}un-Peternell
\cite{BDPP13}.
  Our approach relies on a fine use of quasi-plurisubharmonic envelopes. The
results obtained here will be used in \cite{GL21b} for solving degenerate
complex Monge-Amp\`ere equations on compact Hermitian varieties.
