Transversality conjecture for sections of a general rational elliptic surface

Let $R\to\mathbb P^1$ be a general rational elliptic surface, and let sections mean section curves of this elliptic fibration. Transversality conjecture. Any two sections meet transversely. A special case is stated to have been proved by Ulmer and Urzúa, while the general assertion is used to simplify the analysis of simple telltales and remains open in the supplied context.