Greene–Owens cubiquity conjecture for double branched covers

Let $L$ be a non-split alternating link in $S^3$, let $\Sigma(L)$ denote the double cover of $S^3$ branched along $L$, and let $\Lambda(L)$ be the positive definite lattice associated to $L$.

Greene–Owens cubiquity conjecture. $\Sigma(L)$ bounds a rational homology $4$-ball if and only if $\Lambda(L)$ is cubiquitous.

Greene–Owens proved the forward implication: if $\Sigma(L)$ bounds a rational homology ball, then $\Lambda(L)$ is cubiquitous. The conjecture asserts that cubiquity is also sufficient for the branched double cover to bound a rational homology $4$-ball.