On O'Grady's generalized Franchetta conjecture

We study relative zero cycles on the universal polarized $K3$ surface $X \to $\mathcal{F}$_g$ofdegree$2g - 2$. It was asked by O’Grady if the restriction of any class in ${\mathrm{C}\mathrm{H}}^2(X)$ to a closed fiber $X_s$ is a multiple of the Beauville-Voisin canonical class $c_{X_s}\in {\mathrm{C}\mathrm{H}}_0(X_s)$. Using Mukai models, we give an affirmative answer to this question for $g\le 10$ and $g = 12, 13, 16, 18, 20$.