  The ACC conjecture for local volumes predicts that the set of local volumes
of klt singularities $x\in (X,\Delta)$ satisfies the ACC if the coefficients of
$\Delta$ belong to a DCC set. In this paper, we prove the ACC conjecture for
local volumes under the assumption that the ambient germ is analytically
bounded. We introduce another related conjecture, which predicts the existence
of $\delta$-plt blow-ups of a klt singularity whose local volume has a positive
lower bound. We show that the latter conjecture also holds when the ambient
germ is analytically bounded. Moreover, we prove that both conjectures hold in
dimension 2 as well as for 3-dimensional terminal singularities.
