Higher additive Dehn map conjecture

Assume that $F$ is an algebraically closed field. Let $\beta_n(F)$ be the additive higher Bloch module and ${\cal E}_n(F)$ the Euclidean $F$-scissor congruence group. Higher additive Dehn map conjecture. For every $n$, there exist canonical injective homomorphisms of $F^*$-modules

$$l_n:\beta_n(F)\hookrightarrow{\cal E}_n(F)$$

that commute with the coproduct and volume homomorphisms. This extends the proposed weight-two and weight-three comparisons to all higher weights, while asserting injectivity rather than an isomorphism in general.