The Jacobi-symbol conjecture for the integers $c_p$

Let $p$ be an odd prime. For $\delta\in\{0,1\}$, define

$$D_p^{(\delta)}(w)=\left|\left(\frac{j^2-k^2}{p}\right)+\left(\frac{jk}{p}\right)w\right|_{\delta\leqslant j,k\leqslant(p-1)/2}-\left|\left(\frac{j^2-k^2}{p}\right)\right|_{\delta\leqslant j,k\leqslant(p-1)/2}.$$

For $p\equiv1\pmod4$, let $c_p$ be the positive odd integer whose existence is asserted by the preceding theorem through $D_p^{(0)}(w)=-w((p-1)c_p/2)^2$; for $p\equiv3\pmod4$, let $c_p$ be the positive odd integer whose existence is asserted through $D_p^{(1)}(w)=w(c_p\sum_{k=1}^{(p-1)/2}(\frac{k}{p}))^2$.

Jacobi-symbol conjecture for $c_p$. For any odd prime $p$,

$$\left(\frac{p}{c_p}\right)=1,$$

where $(\frac{p}{c_p})$ is the Jacobi symbol.

The conjecture is based on numerical values of $c_p$ listed in the source. The supplied text gives no resolution or proof, so its status remains open.