Montgomery–Yang conjecture for smooth 4-manifolds with lens-space boundary

Let $M$ be a simply-connected, compact, smooth $4$-manifold with $H_2(M;\mathbb{Z})=\mathbb{Z}$, and suppose that its boundary is a connected sum

$$\partial M=L(p_1,q_1)\#\cdots\#L(p_n,q_n)$$

of lens spaces, where $p_1,\ldots,p_n>1$ are pairwise relatively prime. Montgomery–Yang conjecture. Then $n\leq 3$. This is a smooth reformulation of the original Montgomery–Yang problem via the correspondence with pseudo-free circle actions; the source does not report a complete resolution.