The mod-2 Betti-number characterization of splitting manifolds

Let $M$ be a manifold. Say that $M$ splits when it is divided by every divisor, and let $\beta_1$ be the first Betti number of $M$ with coefficients in $\mathbb{Z}/2\mathbb{Z}$. The splitting characterization conjecture.

$$M\text{ splits}\quad\Longleftrightarrow\quad\beta_1=0.$$

This is proposed as an optimal strengthening of the result that every simply-connected manifold splits; no resolution is given in the source.