The no odd-index handles conjecture for simply connected 4-manifolds

Let $L=L_1\cup L_2$ satisfy the hypotheses of the preceding Kirby-calculus formulation, but impose no restriction on the number of components of $L_2$. If $p+q\ne0$, then $L$ can be transformed by moves (1)--(3) to a similar link with a smaller value of $p+q$. No odd-index handles conjecture. Every closed, simply connected 4-manifold has a handle decomposition without odd-index handles. The source explains that the link formulation is equivalent to this handle-decomposition assertion, whose affirmative resolution would imply SPC4 and, via Property P, the nonexistence of exotic smooth structures on $\mathbb{CP}^2$.