Schoen–Yau–Schick degree-one dominated $S^2$-stability conjecture

Let $X^n$ be an oriented manifold with $n\leq 5$. Call $X^n$ Schoen–Yau–Schick if there exist classes $\alpha_1,\alpha_2,\dots,\alpha_{n-2}\in H^1(X)$ such that

$$[X]\smallfrown\alpha_1\smallfrown\alpha_2\smallfrown\dots\smallfrown\alpha_{n-2}$$

does not lie in the Hurewicz image of $\pi_2(X)$. The degree-one version of dominated $S^2$-stability is the property asserted in the conjecture.

Schoen–Yau–Schick degree-one dominated $S^2$-stability conjecture. Every Schoen–Yau–Schick manifold $X^n$ with $n\leq 5$ has the degree-one version of dominated $S^2$-stability.

The conjecture seeks an $S^2$-stability property for manifolds obstructing positive scalar curvature and is motivated by minimal-hypersurface methods. Its general status is open.