Bloch regulator injectivity conjecture

Let $\mathcal B(\mathbb C)$ be the Bloch group, defined as the kernel of the complex Dehn invariant, and let

$$\rho\colon\mathcal B(\mathbb C)\to\mathbb C/\pi^2\mathbb Q$$

be the Bloch regulator map. Bloch regulator injectivity conjecture. The map $\rho$ is injective. Bloch proved that its image is countable, but injectivity, and hence the countability of $\mathcal B(\mathbb C)$, remains open.