The suspension detection conjecture for maps inducing isomorphisms in homology and fundamental group

Let $A$, $X$, and $Y$ be finite CW-complexes. Suppose that a map

$$f\colon\Sigma X\to\Sigma Y$$

is an isomorphism in $\pi_1$ and homology. Let $g\colon A\to\Sigma X$ be a map such that

$$f\circ g\simeq*.$$

Suspension detection conjecture. Then $g\simeq*$.

This conjecture is proposed as a way to resolve the converse problem for the paper's decomposition theorem. The source emphasizes that an isomorphism in fundamental group and homology does not generally imply a homotopy equivalence, so the assertion is a stronger property specific to maps between suspensions.