  In [6] the author conjectures and partially shows that the Cuntz semigroup
classifies unitary elements of unital AF-algebras. We provide a complete proof
by addressing the existence part of the conjecture, under a mild adjustment of
both domain and codomain of the functor Cu. We also tackle the classification
beyond the AF case and more particularly, we look at unitary elements of what
we call AH$_1$-algebras. We obtain positive progress as far as the existence
part is concern. Nevertheless, we reveal some crucial information needed for
the uniqueness part of the classification that the Cuntz semigroup fails to
capture.
