Full Relative Aspherical Conjecture for positive scalar curvature

Let $Y^{n+k}$ be a compact manifold and let $X^n$ be a codimension-$k$ submanifold with trivial normal bundle. Say that $Y$ is aspherical relative to $X$ when the map $\pi_1(X)\to\pi_1(Y)$ is injective and the maps

$$\pi_i(X)\longrightarrow\pi_i(Y)$$

are isomorphisms for all $i\geq 2$.

Full Relative Aspherical Conjecture. If $n\ne 4$, $Y$ is aspherical relative to $X$, and $X$ admits no metric of positive scalar curvature, then $Y$ admits no metric of positive scalar curvature.

This conjecture proposes a relative topological obstruction to positive scalar curvature in arbitrary codimension, without requiring an a priori assumption on the topology of the ambient manifold. Its general status is open.