Monotonicity of non-pluripolar Monge-Amp\`ere masses

We prove that on a compact K"ahler manifold, the non-pluripolar Monge-Amp`ere mass of a $\theta$-psh function decreases as the singularities increase. This was conjectured by Boucksom-Eyssidieux-Guedj-Zeriahi who proved it under the additional assumption of the functions having small unbounded locus. As a corollary we get a comparison principle for $\theta$-psh functions, analogous to the comparison principle for psh functions due to Bedford-Taylor.