Transitive monodromy conjecture for rational curves on primitive K3 surfaces

Let $S\subset {\mathbb P}^n$ be a primitive K3 surface, and let $d>0$. Consider the family of rational curves in the linear system $|{\mathcal O}_S(d)|$. Transitive monodromy conjecture. Rational curves in $|{\mathcal O}_S(d)|$ have transitive monodromy for every $d>0$. This generalizes Ran's stated result for rational curves of any degree on a quartic surface. The paper presents the assertion as a conjecture, with no resolution supplied here.