The motivic -homotopy contraction conjecture for torus actions

Let $X$ be a projective scheme over a field $k$ with an action of $T=\mathbf{G}_m$, and suppose that $X$ has a $T$-equivariant ample line bundle. Let $Y$ be a $T$-invariant closed subset of $X$ such that every point $x$ in $X$ with

$$\lim_{t \to \infty}(tx)\in Y$$

is in $Y$. Suppose that the fixed point set $Y^T$ is open in $X^T$. Let $U$ be the subset of points $x$ in $X$ such that

$$\lim_{t \to 0}(tx)\in Y.$$

The motivic $\mathbf{A}^1$-homotopy contraction conjecture. The inclusion $Y\to U$ is an $\mathbf{A}^1$-homotopy equivalence, that is, an isomorphism in the $\mathbf{A}^1$-homotopy category $H(k)$.

This conjecture predicts that the attracting locus of a suitable invariant closed subset for a $\mathbf{G}_m$-action is motivically equivalent to that subset. The inclusion is known to be a homotopy equivalence in the classical topology over the complex numbers, but the stated $\mathbf{A}^1$-homotopy equivalence remains conjectural in general.