Blackadar–Handelman density conjecture for dimension functions

Let $A$ be a $C^*$-algebra. Write $DF(A)$ for the set of dimension functions on $A$, and let $LDF(A)$ denote the subset of lower semicontinuous dimension functions. Equip $DF(A)$ with the topology of pointwise convergence.

Blackadar–Handelman density conjecture. The set $LDF(A)$ is dense in $DF(A)$.

This is the first of the two Blackadar–Handelman conjectures on dimension functions. The paper gives an equivalent condition for unital $A$ and proves the conjecture for all unital algebras with finite radius of comparison or almost unperforated Cuntz semigroup, but the statement is not resolved in full generality.