Homological stability conjecture for spaces of polynomial maps

Let $X$ be the variety and $P$ the data defining the spaces of maps $\operatorname{Mor}_{d,P}(\mathbb A^1,X)$, let $\overline K$ be an algebraic closure of the relevant base field, and let $\operatorname{PConf}_m$ denote the ordered configuration space of $m$ points. Assume

$$d\geqslant k-1\geqslant 2,\qquad n>2^k(k-1).$$

Homological stability conjecture. The cohomology group

$$H^{i+2d(n-k)}_c\left(\operatorname{Mor}_{d,P}(\mathbb A^1,X)_{\overline K},\mathbb Q_{\ell}\right)$$

is isomorphic to

$$\bigoplus_{m\geqslant 0}\left(H^{i+m(n-1)}_c\left(\operatorname{PConf}_m,\mathbb Q_{\ell}\right)\otimes H^{n-1}_c\left(X_{\overline K},\mathbb Q_{\ell}\right)^{\otimes m}\otimes \operatorname{sgn}^{n-1}\right)^{S_m}$$

for

$$i>-4\left(\left\lfloor\frac{d}{k-1}\right\rfloor\left(\frac{n}{2^k}-k+1\right)-1\right).$$

This conjecture predicts that the spectral sequence associated with the cohomology of these spaces degenerates on the first page for sufficiently large parameters, yielding a stable description of the cohomology. It is motivated by the role of homological stability in geometric approaches to analytic number theory; the source gives no resolution, so the conjecture remains open.