The pre-Bloch group parity conjecture

Let $F$ be an algebraically closed field. Here $\mathfrak{p}_n(F)$ denotes the primitive part of the homology group associated with $\operatorname{GL}_n(F)$, and $l$ is a prime. The pre-Bloch group parity conjecture.

$$\mathfrak{p}_n(F)\otimes \mathbb{Z}/l=\begin{cases}\mathbb{Z}/l & \text{if $n$ is odd}\\0 & \text{if $n$ is even.}\end{cases}$$

The conjecture predicts the parity pattern of the mod-$l$ primitive homology of general linear groups over algebraically closed fields; by the surrounding discussion it implies the corresponding homology stability statement.