So I'm interested in generalizing the Topological Generalized Poincare Conjecture
To recap the topological generalized Poincare conjecture asks, if we have an $n$-manifold $X$ such that homotopy groups
$$ \pi_1(X), \pi_2(X), ... \pi_N(x) $$
Equal the corresponding homotopy groups
$$ \pi_1(S^n), \pi_2(S^n) ... \pi_N(S^n) $$
Respectively where $S^n$ denotes the $n$-sphere.
Now what I'm curious about is what happens if we don't insist ALL homotopy groups match but only some.
For example for $n=5$ we could ask if $X$ is such that
$$ \pi_1(X) = \pi_1(S^5), \pi_3(X) = \pi_3(S^5), \pi_4(x) = \pi_4(S^5) $$
Then is $X$ necessarily homeomorphic to the $n$-sphere? Of course I would expect if you get rid of sufficiently many constraints the answer should be NO (trivially I believe that not every simply connected $n$-manifold is a sphere), however, it does seem like an interesting question to ask how many homotopy group equivalences do need to present to guarantee that its not always an $n$-sphere and alternatively, given a set of constraints how do you generate a non-spherical counterexample if one exists.
To add a little more commentary, basically in dimension $n$ there are $2^n$ conjectures of this flavor each corresponding to a subset of the $n$ homotopy groups.
The actual question:
Has any one asked this before or worked on any of these weakened topological generalized Poincare conjectures? What sorts of results are known in this space? Are there any interesting examples for example in dimension $n$ preserving $n-q$ (for 'small' $q$) homotopy groups but still not a sphere?