Piezas’s conjecture on universal non-trivial solvability over natural numbers

Establish whether for every natural number a there exists a non-trivial solution to the quartic Diophantine equation A^4 + a B^4 = C^4 + a D^4.

Background

The Brief History section recounts previous computational and theoretical progress and explicitly cites a conjecture attributed to Tito Piezas regarding universal solvability for all a ∈ N.

Subsequent computational work provided extensive evidence but no general proof is presented here, leaving the conjecture as a stated open claim.