  Motivated by the solution of the aspherical conjecture up to dimension 5
[CL20][Gro20], we want to study a relative version of the aspherical
conjecture. We present a natural condition generalizing the model
$X\times\mathbb{T}^k$ to the relative aspherical setting. Such model is closely
related to submanifold obstruction of positive scalar curvature (PSC), and
would be in similar spirit as [HPS15][CRZ23] in codim 2 case. In codim 3 and 4,
we prove results on how 3-manifold obstructs the existence of PSC under our
relative aspherical condition, the proof of which relies on a newly introduced
geometric quantity called the it spherical width. This could be regarded as a
relative version extension of the aspherical conjecture up to dim 5.
