  A family of K3 surfaces $\mathscr{X}\rightarrow B$ has the \emph{Franchetta
property} if the Chow group of 0-cycles on the generic fiber is cyclic. The
generalized Franchetta conjecture proposed by O'Grady asserts that the
universal family $\mathscr{X}_g\rightarrow \mathscr{F}_g$ of polarized K3 of
degree $2g-2$ has the Franchetta property. While this is known only for small
$g$ thanks to \cite{PSY}, we prove that for all $g$ there is a hypersurface in
$ \mathscr{F}_g$ such that the corresponding family has the Franchetta
property.
