Conjecture on Beilinson and Cheeger–Chern classes of universal flat bundles

Let $BGL_n({\mathbb C})^\delta$ be the simplicial classifying space of the discrete group $GL_n({\mathbb C})$, and let $c_p^B$ and $\widehat{c}_p$ be, respectively, the Beilinson and modified universal Chern classes in

$$H^{2p-1}(BGL_n({\mathbb C})^\delta,{\mathbb C}/{\mathbb Z}(p)).$$

Beilinson–Cheeger–Chern class conjecture. For all $p \geq 1$ and $n \in {\mathbb N} \cup\{\infty\}$,

$$c_p^B=\widehat{c}_p \in H^{2p-1}(BGL_n({\mathbb C})^\delta,{\mathbb C}/{\mathbb Z}(p)).$$

If true, this would identify the universal Beilinson Chern class for flat bundles with the modified Cheeger–Chern class and, together with the preceding theorem, imply that it is represented by half the Borel regulator element. The statement is presented as conjectural because the authors were unable to prove the relevant simplicial analogue.