Consider the following question: If $K\subset S^{3}$ is a nontrivial knot. Let $Y$ be the manifold obtained by doing $1/n$-surgery  ($n\geq1$). Is it possible that the instanton Floer homology of $Y$ vanishes?

The answer seems to be no. Because by Gordon-Luecke's theorem, $Y$ is not $S^{3}$. While as far as I know, people don't have any example of an integer homology 3-sphere other than $S^{3}$ having vanishing instanton Floer homology.

My question is: is there actually a rigorous proof of this problem? (for example, using instanton suture Floer homology?)

Consider the following question: Let $K\subset S^{3}$ be a nontrivial knot, and let $Y$ be the manifold obtained by doing $1/n$-surgery ($n\geq1$). Is it possible that the instanton Floer homology of $Y$ vanishes?

The answer seems to be no. Indeed by Gordon-Luecke's theorem, $Y$ is not $S^{3}$, while as far as I know, people don't have any example of an integer homology 3-sphere other than $S^{3}$ having vanishing instanton Floer homology.

My question is: is there a rigorous proof of this statement? (for example, using instanton suture Floer homology?)