Problem 13.24.11D0837305A3CF

Problem 13.24.11.D0837305A3CF For a big nonclosed background $\theta $ on a manifold with the bounded mass property, find criteria for $\underline{\operatorname {vol}}_\theta (P_\theta [u]_{\mathcal I})>0$ in terms of restricted multiplier ideals or divisorial orders. When this lower volume vanishes, which implications between envelope equality and upper-volume equality, conditions (i) and (ii) of proposition 13.24.2, remain valid?