Nonexistence of consecutive powerful triplets around cubes with mixed prime factorizations

A positive integer is powerful if every prime in its factorization occurs with exponent at least two. The Erdős–Mollin–Walsh conjecture asserts that no three consecutive integers are all powerful. Chan (Integers 25 (2025), #A7) excluded triples of the form x³−1=p³y², x³, x³+1=q³z², and She (Integers 25 (2025), #A103) excluded x³−1=p²a³, x³, x³+1=q²b³. This note closes the remaining mixed configurations: there are no consecutive powerful triples with x³−1=p²a³ and x³+1=q³z², nor with x³−1=p³y² and x³+1=q²b³. The case analysis reduces both families to the single equation t⁶+t³+1=3w², which is solved completely via a rank-0 elliptic quotient of a palindromic genus-2 curve. Complementary results: no finite congruence sieve can decide this equation, and Beckon's mod-36 constraint on consecutive powerful triples is refined to arbitrary prime-square moduli. The archive includes the LaTeX source and the complete verification scripts (SymPy, SageMath).