An accepted solution exists. The statement is true.
He88
Heath-Brown, D. R., Ternary quadratic forms and sums of three square-full numbers. Séminaire de Théorie des Nombres, Paris 1986–87, Progr. Math. 75, Birkhäuser Boston (1988), 137-163.
Er76d
Erdős, P., Problems and results on number theoretic properties of consecutive integers and related questions. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44.