Suppose $d \ge 2$ and $S$ is a finite simplicial complex of dimension $2d$, such that
(1) $S$ is simply connected, i.e. $\pi_1 ( S) = 0$, and
(2) all the homology of $S$ is in middle degree, i.e. $\widetilde{H}_i ( S, \mathbb{Z}) = 0,$ unless $i = d$, and
(3) homology $\widetilde{H}_d(S, \mathbb{Z})$ is torsion-free.
Does it necessarily follow that $S$ is homotopy equivalent to a wedge of spheres of dimension $d$?
If $S$ can be shown to be homotopy equivalent to a $d$-dimensional cell complex, for example, this would follow by standard results.