Let $A\subseteq X$ be two finite cell complexes, $\dim X\leq 2n-3$ and let $[(X,A), (S^n, *)]$ be the relative cohomotopy group. There is a natural map $$ \delta: [(X,A),(B^n,S^{n-1})] \to [(X,A), (S^n, *)] $$ induced by $\pi: (B^n, S^{n-1})\to (B^n/S^{n-1}, S^{n-1}/S^{n-1})=(S^n, *)$. The map $\delta$ maps a representant $f$ to $[\pi\circ f]$.
My question is: if $\varphi: (X,A)\to (S^n,*)$ is so that $[\varphi]\in\text{Im}(\delta)$, does there exists an $f: (X,A)\to (B^n, S^{n-1})$ so that $\varphi=\pi\circ f$?
Unfortunately, we cannot in general lift a homotopy in $[(X,A), (S^n, *)]$ to a homotopy in $[(X,A), (B^n, S^{n-1})]$, because $\pi$ is not a fibration. However, I believe that the statement still holds in the stable dimension range..