Distinct Frobenius-trace tuples for the shifted elliptic curves

Let $p$ be prime and let $u\in\mathbb F_p$ be a parameter for which the 15 elliptic curves obtained from the five elliptic-curve factors and the shifts $u\mapsto u-22$ and $u\mapsto u-30$ are nonsingular. Their Frobenius traces form an ordered $15$-tuple. Distinct-trace-tuple conjecture. For all primes, generic $u\in\mathbb F_p$ yield distinct $15$-tuples of Frobenius traces. The conjecture would make these tuples effective invariants for distinguishing generic parameters in the $D_6$ family, after accounting for the repeated parametrization by the choice of a rational Weierstrass point. Its resolution is not specified in the source.