Signature-zero prime-determinant linking-form conjecture

Let $K$ be a knot, let $\sg(K)$ denote its signature, let $D=\dt(K)$ be its prime determinant, let $H_1$ denote the first homology group of the double branched cover of $K$, and let $\lm$ denote its linking form. Signature-zero linking-form conjecture. If $K$ has prime determinant $D$ and signature zero, then there is an element $g\in H_1$ such that

$$\lm(g,g)=\pm\frac{2}{D}.$$

The claim is known for signature $2$ in the related discussion, while the signature-zero case is presented as an empirically supported open problem.