Kollár's conjecture on Brieskorn spheres and integer homology balls

For integers $a_1,\ldots,a_n$, let $\Sigma(a_1,\ldots,a_n)$ denote the corresponding Brieskorn sphere. A smooth integer homology ball is a smooth $4$-manifold with the same integer homology as the $4$-ball.

Kollár's conjecture. No Brieskorn sphere $\Sigma(a_1,\ldots,a_n)$ with $n\geq 4$ is the boundary of a smooth integer homology ball.

The conjecture is connected with the Montgomery–Yang problem on circle actions on $S^5$ and with the Bogomolov–Miyaoka–Yau inequality. The source describes it as longstanding and gives no resolution.