Suppose $r\geq 0$ is a rank attainable by infinitely many elliptic curves over $\mathbb{Q}$. Let $T$ be one of the fifteen finite abelian groups in Mazur's theorem.

Is there an elliptic curve $E/\mathbb{Q}$ such that $E(\mathbb{Q})\approx \mathbb{Z}^r\times T$?