Franchetta conjecture for families with motivic multiplicativity

Let $X$ and $B$ be smooth connected varieties, and let $f:X\to B$ be a smooth projective morphism such that a very general fiber of $f$ admits motivic $0$-multiplicativity. Assume that, for a very general point $b\in B$, the monodromy-invariant sub-Hodge structure $H^*(X_b,\mathbb{Q})^{\pi_1(B,b)}$ consists of Hodge classes. Franchetta conjecture for motivically multiplicative families. For every integer $N\geq 1$ and every cycle class $\Gamma\in\operatorname{CH}^*(X_{/B}^N)$, the restriction $\Gamma|_{X_b^N}$ vanishes for every closed point $b\in B$ whenever it is homologically trivial for a very general point $b\in B$. This generalizes the Franchetta property known in the source for certain universal families and is proposed as a generalization of the hyper-Kähler case; its status is not resolved there.