The knot-independence conjecture for the essential instanton Floer summand

Let $(\widehat{Y},\widehat{K})$ satisfy the hypotheses of the source's torsion spin$^c$ decomposition theorem, and let $\widehat{K}'\subset\widehat{Y}$ be another knot satisfying the analogous conditions. Assume

$$[\widehat{K}]=[\widehat{K}']\in H_1(\widehat{Y}).$$

Let $\mathcal{I}_+(\widehat{Y},\widehat{K})$ denote the essential component associated with the pair. Knot-independence conjecture. There is a grading-preserving isomorphism

$$\mathcal{I}_+(\widehat{Y},\widehat{K})\cong\mathcal{I}_+(\widehat{Y},\widehat{K}')$$

up to a $\mathbb{Z}_q$ grading shift. The conjecture asserts that the decomposition of $I^{\sharp}(\widehat{Y})$ is independent of the choice of knot representing the relevant homology class. The source says that this independence is not known.