  We show that an old conjecture of A.A. Suslin characterizing the image of a
Hurewicz map from Quillen K-theory in degree $n$ to Milnor K-theory in degree
$n$ admits an interpretation in terms of unstable ${\mathbb A}^1$-homotopy
sheaves of the general linear group. Using this identification, we establish
Suslin's conjecture in degree $5$ for infinite fields having characteristic
unequal to $2$ or $3$. We do this by linking the relevant unstable ${\mathbb
A}^1$-homotopy sheaf of the general linear group to the stable ${\mathbb
A}^1$-homotopy of motivic spheres.
