Vanishing conjecture for the second-page spectral-sequence terms

Let $F$ be an infinite field and let $M_\bullet$ be the exact complex of $\operatorname{GL}_n(F)$-modules used to construct the first-quadrant spectral sequence $E^r_{p,q}(n,\mathbb Z)$ converging to zero. Vanishing conjecture for $E^2_{i,n-i+2}(n,\mathbb Z)$. For $n\geq3$ and every $3\leq i\leq n+2$,

$$E^2_{i,n-i+2}(n,\mathbb Z)=0.$$

In particular, $E^2_{1,n}(n,\mathbb Z)=0$, and the differential $d^2_{2,n}(n,\mathbb Z):E^2_{2,n}(n,\mathbb Z)\to E^2_{0,n+1}(n,\mathbb Z)$ is surjective. The statement is presented as a conjecture, and no resolution is supplied in the source.