Morel's exact sequence conjecture for motivic homotopy sheaves

Let $k$ be a base field, let $n$ and $i$ be integers with $n\geq 4$ and $i\geq 0$, and write $S^{(n-1+i)+n\alpha}$ for the corresponding motivic sphere. Let $\mathbf K^{\mathrm{M}}_{n+2}/24$ denote Milnor $K$-theory modulo $24$, let $\bm{\pi}_{n}^{{\mathbb A}^1}$ denote the $n$th $\mathbb A^1$-homotopy sheaf, and let $\mathbf{GW}^n_{n+1}$ denote the indicated Grothendieck–Witt sheaf. Morel's conjecture. For every pair of integers $n\geq 4$ and $i\geq 0$, there is an exact sequence of the form

$$\mathbf K^{\mathrm{M}}_{n+2}/24\longrightarrow \bm{\pi}_{n}^{{\mathbb A}^1}(S^{(n-1+i)+n\alpha})\longrightarrow \mathbf{GW}^n_{n+1};$$

the right-hand map becomes an epimorphism after $(n-3)$-fold contraction, and the sequence becomes short exact after $n$-fold contraction. This refines Morel's conjecture on the structure of these motivic homotopy sheaves; the source states that Morel's conjecture has been verified in various situations, while the displayed assertion is obtained here under the hypotheses and results developed in the paper.