  We formally introduce the concept of localizing the Elliott conjecture at a
given strongly self-absorbing C*-algebra $D$; we also explain how the known
classification theorems for nuclear C*-algebras fit into this concept. As a new
result in this direction, we employ recent results of Lin to show that (under a
mild K-theoretic condition) the class of separable, unital, simple C*-algebras
with locally finite decomposition rank and UCT, and for which projections
separate traces, satisfies the Elliott conjecture localized at the Jiang-Su
algebra Z.
  Our main result is formulated in a more general way; this allows us to
outline a strategy to possibly remove the trace space condition as well as the
K-theory restriction entirely. When regarding both our result and the recent
classification theorem of Elliott, Gong and Li as generalizations of the real
rank zero case, the two approaches are perpendicular in a certain sense. The
strategy to attack the general case aims at combining these two approaches.
  Our classification theorem covers simple ASH algebras for which projections
separate traces (and the K-groups of which have finitely generated torsion
part); it does, however, not at all depend on an inductive limit structure.
Also, in the monotracial case it does not rely on the existence or absence of
projections in any way. In fact, it is the first such result which, in a
natural way, covers all known unital, separable, simple, nuclear and stably
finite C*-algebras of real rank zero (the K-groups of which have finitely
generated torsion part) as well as the (projectionless) Jiang-Su algebra
itself.
