Rational points on quadratic elliptic surfaces

We consider elliptic surfaces whose coefficients are degree $2$ polynomials in a variable $T$. It was recently shown that for infinitely many rational values of $T$ the resulting elliptic curves have rank at least $1$. In this article, we prove that the Mordell-Weil rank of each such elliptic surface is at most $6$ over $\mathbb{Q}$. In fact, we show that the Mordell-Weil rank of these elliptic surfaces is controlled by the number of zeros of a certain polynomial over $\mathbb{Q}$.