Reducibility conjecture for Severi varieties on rational elliptic surfaces

Let $R$ be a rational elliptic surface, and let $V(R,L)$ denote the genus $0$ Severi variety associated to a line bundle $L$ on $R$. Reducibility conjecture. There exist line bundles $L$ for which $V(R,L)$ has dimension $1$ and is reducible. This would show that the irreducibility theorem for positive-dimensional Severi varieties on del Pezzo surfaces is sharp and does not extend to rational elliptic surfaces. The conjecture is presented as an expected failure of irreducibility in the rational elliptic-surface setting.