Exactness conjecture for the homology sequence of general linear groups

Let $k$ be a field, let $F^\ast$ denote the multiplicative group of a field $F$, and let $H_n(-,k)$ denote homology with coefficients in $k$. For $n\geq 3$ such that $(n-1)!\in k^\ast$, consider the sequence

$$H_n(F^\ast{}^2\times \operatorname{GL}_{n-2},k)\xrightarrow{\beta_2^{(n)}}H_n(F^\ast\times \operatorname{GL}_{n-1},k)\xrightarrow{\beta_1^{(n)}}H_n(\operatorname{GL}_n,k)\longrightarrow 0.$$

Exactness conjecture. This sequence is exact.

The conjecture proposes a homological stability-type exact sequence relating the homology of general linear groups over an infinite field. The supplied text gives no evidence that the conjecture has been resolved.