Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ denote the Legendre symbol. Motivated by the evaluation of the determinants
$$\det\left[\left(\frac{j+k}p\right)\right]_{1\le j,k\le(p-1)/2}\ \ \text{and}\ \ \det\left[\left(\frac{j-k}p\right)\right]_{1\le j,k\le(p-1)/2}$$
(cf. <a href="https://mathoverflow.net/questions/470324">Question 470324</a>), I have formulated the following conjecture based on my computation via Mathematica.

**Conjecture.** Let $p>3$ be a prime. 

(i) We have 
$$\begin{aligned}&\ \left|\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)\right|_{1\le j,k\le (p-1)/2}
\\=&\ \begin{cases}(\frac 2p)p^{(p-5)/4}&\text{if}\ p\equiv1\pmod4,
\\(-1)^{(h(-p)-1)/2}p^{(p-3)/4}&\text{if}\ p\equiv3\pmod4,
\end{cases}
\end{aligned}\tag{1}$$
where $h(-p)$ denotes the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$.
When $p\equiv1\pmod4$, we also have
$$\left|\left(\frac{j+k}p\right)-\left(\frac{j-k}p\right)\right|_{1\le j,k\le (p-1)/2}
=(-p)^{(p-1)/4}.\tag{2}
$$

(ii) We have
$$\begin{aligned}&\ \det\left[x+\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)\right]_{0\le j,k\le(p-1)/2}
\\ =&\ \begin{cases}(\frac 2p)p^{(p+3)/4}x&\text{if}\ p\equiv1\pmod4,
\\(-1)^{(h(-p)-1)/2}p^{(p+1)/4}x&\text{if}\ p\equiv3\pmod4.
\end{cases}\end{aligned}\tag{3}$$

**QUESTION**. What method or tool can be used to study the above problem? Are the quadratic Gauss sums helpful?

Your comments are welcome!