Cohen–Sun–Vsemirnov's integral-square conjecture for S(1,p)

Let $p$ be a prime represented as

$$p=a^2+4b^2,$$

where $a,b$ are integers and $a\equiv1\pmod4$. Let

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

Cohen–Sun–Vsemirnov's conjecture. The number $S(1,p)/a$ is an integral square.

The conjecture is a refinement of known congruence information about Legendre-symbol determinants. It is confirmed in the paper: the stated $d=1$ case follows from the theorem asserting that $\varepsilon(d)S(d,p)/a$ is an integral square.