The existence of a simple nuclear algebra with identical scaled projection semigroups but no matrix isomorphism

Let $A$ be a simple, unital, separable, and nuclear $C^*$-algebra of stable rank one. Write $V(A)$ for its Murray–von Neumann semigroup of projections and $[1_A]$ for the class of its unit. Counterexample to Elliott's classification conjecture. There exists such an algebra $A$ for which

$$A \ncong \mathrm{M}_n(A) \quad\text{for some } n\in\mathbb{N},$$

yet

$$(V(A),[1_A])\cong (V(\mathrm{M}_n(A)),[1_{\mathrm{M}_n(A)}])\cong (\mathbb{Q}^+,1).$$

This provides a particularly strong counterexample to Elliott's classification conjecture for simple, separable, nuclear $C^*$-algebras, since the scaled ordered projection semigroup does not distinguish $A$ from a matrix algebra over $A$.