The concordance-invariance conjecture for modulo-4 reduced knot-homology ranks

Let $K\subset S^3$ be a knot, let $\mathcal{A}$ be either $\mathbb{F}_p$ or $\mathbb{Q}$, and let $\mathcal{C}$ denote the knot concordance group. Concordance-invariance conjecture. The quantities

$$\operatorname{rk}\widehat{\operatorname{HFK}}(K)\pmod 4$$

and

$$\operatorname{rk}\widetilde{\operatorname{Kh}}_{\mathcal{A}}(K)\pmod 4$$

are concordance invariants; equivalently, they define homomorphisms from $\mathcal{C}$ to $(\mathbb{Z}/4\mathbb{Z})^*$ by sending $K$ to the corresponding residue class.

The paper states that this conjecture is equivalent to the modulo-$4$ ribbon-knot rank conjecture. It would provide smooth sliceness obstructions and potentially detect elements of order $2$ in the concordance group; the general assertion remains open.