The real rank versus completely positive rank conjecture for C*-algebras

Let $A$ be a $C^*$-algebra. Write $\mathrm{rr}\,A$ for its real rank and $\mathrm{cpr}\,A$ for its completely positive rank.

Real-rank inequality conjecture. For every $C^*$-algebra $A$, one has

$$\mathrm{rr}\,A \leq \mathrm{cpr}\,A.$$

The inequality is motivated by examples in which real rank is at most completely positive rank, together with the expected agreement of completely positive rank with covering dimension for commutative algebras. Its resolution is not specified in the supplied text.