Let $f: X \to \Delta \subset \mathbb{C}$ be a projective morphism of a complex manifold to a small disc, smooth away from 0, and such that $Y=f^{-1}(0)=\sum E_i$ is a strictly normal crossing divisor, and let $p: \tilde{\Delta^*} \to \Delta^*=\Delta \setminus \{0\}$ be the universal cover. Denote $X_\infty = X \times_\Delta \tilde{\Delta^*}$ and $E_I=\cap_{i \in I} E_i$. Let $Z \subset X$ be a complex analytic subset, flat over $\Delta$ and meeting the strata of $Y$ transversely.
The Steenbrink's weight monodromy spectral sequence computes $\mathrm{gr}^W H^*(X_\infty)$, degenerates at $E_2$ and the terms of its first page ${}_W E_1^{*,*}$ are direct sums of the cohomologies of starata $E_I$. What is the relationship between the cycle classes of $Z \cap E_I$ in $H^*(E_J)$ (for varying multiindices $I, J$) and the cycle class of $p^{-1}(Z)$ in $H^*(X_\infty)$? Given $Z$ can one write down an element of ${}_W E_2^{*,*}$ so that its image in $\mathrm{gr}^W H^*(X_\infty)$ coincides with the image of the cycle class of $p^{-1}(Z)$?