Silverman's rank conjecture for elliptic curves in one-parameter families

Let $E$ be an elliptic curve over $\mathbb{Q}(t)$, let $E(\mathbb{A}^1)$ denote the elliptic curve over the function field $\mathbb{Q}(t)$, and let $E(t)$ denote its specialization at $t\in\mathbb{Q}$. Order rational numbers $t$ by height. Silverman's conjecture. For almost all $t\in\mathbb{Q}$, one has

$$\operatorname{rank} E(\mathbb{A}^1)\leqslant \operatorname{rank} E(t)\leqslant 1+\operatorname{rank} E(\mathbb{A}^1).$$

This is presented as a natural analogue of Goldfeld's conjecture for quadratic twists. The supplied text does not state a resolution, so the conjecture is recorded as open.