Conformal-flat connected-sum conjecture for manifolds with vanishing Stiefel–Whitney classes

Let $M^n$ be a closed connected smooth $n$-manifold whose Stiefel–Whitney classes all vanish. A Riemannian metric is conformally flat when its Weyl tensor vanishes. Conformal-flat connected-sum conjecture. There exists a smooth closed orientable $n$-manifold $N$ such that $M\mathbin{\#}N$ admits a conformally flat Riemannian metric. The claim generalizes the preceding four-dimensional existence theorem and parallels known constructions in dimension three; the supplied text does not establish the general statement or indicate whether it has been resolved.