Nonempty positive-rank fibers conjecture for the elliptic surface defined by H

Let $a,b,c,d,e,f\binom{}{} \in\mathbb{Z}$ and consider the elliptic surface

$$\mathcal{E}:\; H(U,V,t)=\sum_{i+j\leq 3}a_{i,j}U^iV^j=0,$$

where the coefficients $a_{i,j}$ are given by (ais). For $t\in\mathbb{Q}$, let $\mathcal{E}_t$ denote the corresponding fiber. Positive-rank fibers conjecture. The set

$$S=\{t\in\mathbb{Q}:\; \mathcal{E}_t\text{ is an elliptic curve and has positive rank}\}$$

is nonempty.

The conjecture asserts the existence of a rational parameter yielding an elliptic fiber of positive Mordell–Weil rank. The source presents this as a conjecture motivated by computations; no resolution is supplied here.