Rationally connectedness and O'Grady's generalized Franchetta conjecture for K3 surfaces

O’Grady’s generalized Franchetta conjecture asks whether any codimension two cycle on the universal polarized K3 surfaces restricts to a multiple of the Beauville–Voisin class on a given K3 surface. We apply Bergeron–Li’s result on the cohomology of universal K3 surfaces to give an affirmative answer to this conjecture when the universal K3 surface is rationally connected, including the case of genus 22.