*QUESTION*. Is my following conjecture true? If true, how to prove it?

**Conjecture**. Let $p$ be a prime with $p\equiv5\pmod 6$, and define the matrices $A_p$ and $B_p$ by
$$A_p:=\left[\frac1{i^2-ij+j^2}\right]_{1\le i,j\le p-1}
\ \ \text{and}\ \ B_p=\left[\frac1{i^2-ij+j^2}\right]_{1\le i,j\le (p-1)/2}.$$
Then $$\left(\frac{\det A_p}p\right)=\left(\frac 2p\right)\ \ \text{and}\ \ \text{ord}_p(\det B_p)=\frac{p+1}6,$$
where $(\frac{\cdot}p)$ is the Legendre symbol and $\text{ord}_p(x)$ is the $p$-adic order (or valuation) of $x$.

I formulated the conjecture in 2013 and checked it via Mathematica. It appeared in Remark 1.3 of my recent paper [Finite Fields Appl. 56(2019), 285-307].

I think some of you might be able to prove the conjecture. Your comments are welcome!