Sun's infinitude conjecture for the determinants $[c,d]_p$

Let $p$ be an odd prime and let $c,d$ be integers. Define

$$[c,d]_p:=\det\left[\left(\frac{i^2+cij+dj^2}{p}\right)\right]_{0\le i,j\le p-1}.$$

Sun's conjecture. If $d$ is non-zero and $c^2-4d\ne0$, then $[c,d]_p=0$ for infinitely many primes $p$.

The paper later states and proves a stronger result, so this conjecture is solved.