Let $n\in\mathbb{N}$ Is the volume functional continuous on the set of isometry classes of compact riemannian $n$-manifolds with volume $\geq \varepsilon$\ (with respect to Gromov--Hausdorff distance)?

Without the volume bound, a collapsing torus gives a counterexample. But it seems that this is the only singularity. There are a number of weird results on the semi-continuity of the volume functional. For instance, every metric on $S^3$ is the limit of a metric with volume converging to $0$. I don't know what happens to the curvature of these metrics.