Euclidean Dehn complex cohomology conjecture

Let $F$ be an algebraically closed field, and let ${\cal E}_{(n)}^{\ast}(F)$ be the degree-$n$ Euclidean Dehn complex of Euclidean scissor congruence groups. For an $F$-vector space $V$, write $V<n>$ for the $F^*$-module on which $f\in F^*$ acts by multiplication by $f^{2n+1}$. Let $\Omega^j_{F/\mathbb{Q}}$ denote the space of Kähler $j$-forms over $\mathbb{Q}$. Euclidean Dehn complex conjecture. If $F$ is algebraically closed, then there are canonical isomorphisms of $F^*$-modules

$$H^i({\cal E}_{(n)}^{\ast}(F))=\Omega^{i-1}_{F/\mathbb{Q}}<n>.$$

The same assertion is proposed for the classical case $F=\mathbb{R}$. This predicts that the Euclidean Dehn complex computes differential forms, extending the expected relationship between scissor congruence groups and additive or infinitesimal motivic structures.