Purely classical proof of the representation congruence

Prove the congruence \((r_f(pq)-r_g(pq))/32=(r_a(pq)-h(-pq))/16\equiv\Pi(p,q)\pmod 2\) using only composition and reciprocity of quadratic forms, thereby giving a purely classical explanation of the congruence.

Background

The paper establishes the congruence for primes $p$ and $q$ satisfying $p\equiv q\equiv1\pmod 8$ and $(q/p)=1$ by identifying the normalized ternary quadratic-form representation defect with the Pfaffian of the Cassels pairing on the pure $2$-Selmer group of the congruent number curve $E_{pq}:y^2=x^3-p^2q^2x$. The resulting formula is $$(r_f(pq)-r_g(pq))/32=(r_a(pq)-h(-pq))/16\equiv\Pi(p,q)\pmod2$$.

The unresolved issue is whether this congruence can be derived without elliptic-curve descent and Cassels-pairing methods, relying instead solely on classical composition and reciprocity properties of quadratic forms. The paper notes that such a derivation would provide a purely classical interpretation of the congruence; it also suggests that the reciprocity constraints arising from the alternating Cassels pairing might be expressible through rational quartic and octic reciprocity.