Mori's congruence conjecture for the caliber of real quadratic fields

For prime numbers $p$ and $q$ with $p<q$ and $p\equiv q\equiv 1\pmod 4$, let $x_p,y_p,x_q,y_q$ be the unique positive integers satisfying

$$p=x_p^2+y_p^2,\qquad q=x_q^2+y_q^2,$$

with $0<x_p<y_p$ and $0<x_q<y_q$. Write $\kappa^+(pq)$ for the positive caliber of the real quadratic field associated with $pq$, and let $\left(\frac{q}{p}\right)$ denote the Legendre symbol. Mori's congruence conjecture. If $x_p\not\equiv x_q\pmod 2$, then

$$\kappa^+(pq)\equiv 1-(-1)^{x_p}\left(\frac{q}{p}\right)\pmod 4.$$

This conjecture gives a mod-$4$ formula for the positive caliber in terms of the representations of the primes as sums of two squares and a quadratic-residue symbol. It is attributed in the source to Mori; its resolution is not specified in the supplied material.