Conjecture on linking forms and Jones–Rong polynomial values for knots with signature four

Let $K$ be a knot, let $\sg(K)$ denote its signature, let $H_1$ denote the first homology group of the double branched cover of $K$, and let $Q_K$ be the associated polynomial value used in the paper. A finite abelian group is a double when it is isomorphic to a direct sum of two equal cyclic groups. Linking-form and polynomial-value conjecture. There are no knots with $\sg(K)=4$ and cyclic $H_1$ of order a prime square. If $\sg(K)=4$ and $H_1\cong\mathbb Z_5\oplus\mathbb Z_5$, then

$$Q_K\left(\frac{\sqrt{5}-1}{2}\right)=-5$$

rather than $+5$. These assertions concern unexplained empirical phenomena connecting the signature, the homology and linking form of the double branched cover, and the polynomial value; the source presents them as problems supported by strong empirical evidence, with no resolution supplied here.