On O'Grady's generalized Franchetta conjecture

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