Relation between the equivariant instanton and Heegaard Floer slice invariants

Let $K$ be a knot in $S^3$. Let $q_M(K)$ and $q_M^\dagger(K)$ be the invariants defined from equivariant singular instanton Floer theory, let $q_\tau(K)$ be Hendricks–Lipshitz–Sarkar's invariant from equivariant Heegaard Floer homology, and let $\sigma(K)$ denote the signature of $K$. The $q_M$–$q_\tau$ conjecture. For every knot $K$ in $S^3$,

$$q_\tau(K)=2q_M^\dagger(K),$$

or equivalently,

$$q_\tau(K)=-2q_M(K)-\frac{3}{2}\sigma(K).$$

This conjecture predicts a precise relationship between invariants arising from equivariant instanton and Heegaard Floer theories; the source provides no resolution evidence.