Ref proof for equidistribution of roots of $x^2 \equiv 1$ (mod $p$), as $p$ varies over primes?

Question

This question is about equidistribution of roots of quadratic congruences of the form $x^2 \equiv n$ (mod $p$), which we note always have determinant $D = b^2 - 4ac = 4n$. This is for a fixed $n$ while $p$ varies over primes.

The equidistribution papers typically referenced in this area have significant constraints:

• [DFI 1995] proves equidistribution for all quadratic congruences with negative discriminant D.
• [Toth 2000] extends the proof to quadratics with positive discriminants, but D must not be a square.
• [DFI 2012] extends the proof to quadratics with positive fundamental discriminants, then sketches how to address negative fundamental discriminants. They say, “Extensions to non-fundamental discriminants are also possible”, but they do not say how to do so. This means the positive values of n are restricted to {2,3,6,7,10,11,etc}, to produce a fundamental discriminant.

So we see that n=1, D=4 has a positive discriminant that is both non-fundamental and is a square. Thus the above papers do not seem to prove equidistribution for the roots of $x^2 \equiv 1$ (mod $p$).

Yet, respected authors including [FI 2010] and [SSZ 2020] claim that we have established equidistribution for ALL simple quadratic congruences $x^2 \equiv n$ (mod $p$) over primes, without exceptions. If so, there must be a proof of equidistribution for non-fundamental and/or square positive determinants. I have not been able to search out such a proof.

Can someone please provide a reference to an authoritative proof for this, hopefully with best bounds for the discrepancy?

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References
[DFI 1995] Duke, W., J. B. Friedlander, and H. Iwaniec. “Equidistribution of roots of a quadratic congruence to prime moduli.” Annals of Mathematics 141 (1995): 423–41.
[Toth 2000] A. T ́oth, ‘Roots of quadratic congruences’, Int. Math. Res. Not., 14 (2000), 719–739.
[DFI 2012] W. Duke, J. Friedlander and H. Iwaniec, ‘Weyl sums for quadratic roots’, Int. Math. Res. Not. (2012), 2493 - 2549.
[FI 2010] Friedlander, J. B. and H. Iwaniec. Opera de Cribro. Colloquium Publications 57. Providence, RI: American Mathematical Society, 2010; ref. Theorem 19.2
[SSZ 2020] Shkredov, I.D., Shparlinski, I.E., & Zaharescu, A. (2020). On the distribution of modular square roots of primes. arXiv: Number Theory.