Sun's quadratic-character conjecture for $D_p^{(3)}$

For positive integers $m,n$ with $n$ odd, define

$$D_n^{(m)}=\det\left[(j^2-k^2)^m\left(\frac{j^2-k^2}n\right)\right]_{1\leqslant j,k\leqslant(n-1)/2}.$$

For primes $p\equiv1\pmod4$, this determinant is an integer square, so choose a square root $\sqrt{D_p^{(3)}}$. Sun's quadratic-character conjecture. For every prime $p\equiv1\pmod4$,

$$\left(\frac{\sqrt{D_p^{(3)}}}p\right)=(-1)^{\left|\left\{0<k<\frac p4:\left(\frac kp\right)=-1\right\}\right|}\left(\frac p{4+(-1)^{(p-1)/4}}\right).$$

The source reports verification for all primes $p<1000$ with $p\equiv1\pmod4$, but gives no proof or resolution.