Asok–Fasel conjecture on the th -homotopy sheaf of punctured affine space

Let $k$ be a perfect field with ${\rm char}(k)\ne 2$, and let $d$ be the dimension of a $k$-scheme $X$. Write $\mathbf{K}^{\mathrm{M}}_{d+2}/24$ for the quotient of the Milnor $K$-theory sheaf by multiplication by $24$, $\pi_{d}^{\mathbb{A}^1}(\mathbb{A}^{d}\setminus 0)$ for the $d$th $\mathbb{A}^1$-homotopy sheaf of punctured affine $d$-space, and $\mathbf{GW}^d_{d+1}$ for the indicated Grothendieck–Witt sheaf. The cohomology groups are Nisnevich cohomology groups on $X$. Asok–Fasel conjecture. There is an exact sequence in $\mathcal{A}b_{k}^{\mathbb{A}^1}$

$$\mathbf{K}^{\mathrm{M}}_{d+2}/24\to\pi_{d}^{\mathbb{A}^1}(\mathbb{A}^{d}\setminus 0)\to\mathbf{GW}^d_{d+1}\to0.$$

Consequently, if $X$ is a $k$-scheme of dimension $d$, there is an exact sequence

$$\mathrm{H}^d(X;\mathbf{K}^{\mathrm{M}}_{d+2}/24)\to\mathrm{H}^d(X;\pi_{d}^{\mathbb{A}^1}(\mathbb{A}^{d}\setminus 0))\to\mathrm{H}^d(X;\mathbf{GW}^d_{d+1})\to0.$$

The conjecture describes the first unknown part of the relevant unstable $\mathbb{A}^1$-homotopy sheaf and, through the induced cohomology sequence, is used to study obstruction groups for vector bundles; the supplied text gives no evidence that it has been resolved.