Nodal and transverse intersection conjecture for trigonal K3 surfaces

Let $S$ be a surface given in ${\bf{TK1}}$, ${\bf{TK2}}$, or ${\bf{TK3}}$. Let $F$ and $C$ denote the divisor classes used in these trigonal K3 surface constructions, and let $l$ be the integer occurring in the linear system $|C+lF|$. Nodal and transverse intersection conjecture. Every irreducible rational curve in $|F|$ is nodal, and it intersects transversely with every irreducible rational curve in $|C+lF|$. This is an auxiliary assumption used to analyze limiting rational curves on degenerations to trigonal K3 surfaces; the paper gives no proof of it.