Let $X$ be a $k$-dimensional triangulated manifold and $A:=\partial X$ be its boundary. Let $f:X\to R^d$ be a continuous function such that $g:=f|_{A\cup X^{(i-1)}}$ avoids zero and let $z_g\in Z^{i} \big(X,A; \pi_{i-1} (R^d\setminus\{0\})\big)$ be the obstruction to extending $g$ to $A\cup X^{(i)}\to R^d\setminus\{0\}$. Let $\pi_{i-1}$ denote the $(i-1)$st homotopy group of $S^{d-1}\simeq R^d\setminus\{0\}$. Is it true that the image of the map $H_{k-i}(f^{-1}(0); \pi_{i-1} )\to H_{k-i}(X;\pi_{i-1})$ induced by the inclusion contains the Poincare dual of $[z_g]$?

This is known to be true for $i=d$. If it was true, it would help us to get computable topological invariants of the zero sets of $f$ stable under perturbations of $f$ within a given range in $\|\cdot\|_\infty$-norm.