Motivic spheres and the image of the Suslin--Hurewicz map

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 Suslins 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.