O'Grady's generalized Franchetta conjecture for polarized K3 surfaces

For each $g\ge 3$, let $\mathcal F_g'$ be the smooth dense open moduli space of primitively polarized K3 surfaces of degree $2g-2$ with trivial automorphism groups, and let

$$\pi:\mathcal S_g'\to\mathcal F_g'$$

be the universal K3 surface. For a closed point $x\in\mathcal F_g'$, write $S_x:=\pi^{-1}(x)$, and let $o_{S_x}$ denote its Beauville–Voisin class in $\operatorname{CH}^2(S_x)$. O'Grady's generalized Franchetta conjecture. For every class $a\in\operatorname{CH}^2(\mathcal S_g')$, the restriction $a|_{S_x}$ is a multiple of $o_{S_x}$.

The conjecture asks whether codimension-two cycles on the universal polarized K3 surface restrict on every fiber to multiples of the canonical Beauville–Voisin zero-cycle. The abstract states that the conjecture has an affirmative answer in genus $11$, while the assertion for general $g\ge 3$ is not established here.