Let $\pi \colon X \to S$ be a smooth projective morphism of algebraic varieties, say over $\mathbf C$. By Deligne's argument ("Théorème de Lefschetz...", 1968), there is for each $i$ an injection $$ \newcommand{\Q}{\mathbf{Q}} R^i \pi_\ast\Q[-i] \hookrightarrow R\pi_\ast\Q,$$ such that the direct sum of all these gives a quasi-isomorphism between $R\pi_\ast\Q$ and its cohomology.

**Question:** Can this construction be made compatible with cup-product? That is, can one choose these injections so that the diagram
$$ \begin{matrix}
R^i\pi_\ast\Q[-i] \otimes R^j\pi_\ast\Q[-j] & \to & R^{i+j}\pi_\ast\Q[-i-j] \\\\
\downarrow & & \downarrow \\\\
R\pi_\ast\Q \otimes R\pi_\ast\Q & \to & R\pi_\ast\Q
\end{matrix}$$
commutes? If not, can one write down an obstruction?

The question is motivated by the fact that the fibers are compact Kähler manifolds, hence formal by Deligne-Griffiths-Morgan-Sullivan. So on each fiber $X_s$, we have a quasi-isomorphism with the cohomology when both are considered as dg *algebras*. Hence an affirmative answer would be a version of DGMS's result which is moreover valid in families.