  We investigate the Hausdorff measure and content on a class of quasi
self-similar sets that include, for example, graph-directed and sub
self-similar and self-conformal sets. We show that any Hausdorff measurable
subset of such a set has comparable Hausdorff measure and Hausdorff content. In
particular, this proves that graph-directed and sub self-conformal sets with
positive Hausdorff measure are Ahlfors regular, irrespective of separation
conditions. When restricting to self-conformal subsets of the real line with
Hausdorff dimension strictly less than one, we additionally show that the weak
separation condition is equivalent to Ahlfors regularity and its failure
implies full Assouad dimension. In fact, we resolve a self-conformal extension
of the dimension drop conjecture for self-conformal sets with positive
Hausdorff measure by showing that its Hausdorff dimension falls below the
expected value if and only if there are exact overlaps.
