Strong 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 weakly aspherical relative to $X$ when the map $\pi_1(X)\to\pi_1(Y)$ is injective, the maps

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

are isomorphisms for $i=2,\dots,k-1$, and the map

$$\pi_k(X)\longrightarrow\pi_k(Y)$$

is surjective.

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

This is a weaker homotopy-theoretic hypothesis than full relative asphericity and is intended to extend positive-scalar-curvature obstructions to broader relative settings. Its general status is open.