Setup:
Let $R$ be a finitely generated subring of $\mathbb{C}$. Let $X \rightarrow \mathbb{A}^1_R$ be a proper morphism of $R$-varieties, smooth except over a rational point $s \in \mathbb{A}^1_R$ that has finite residue field $\kappa$.
Let $K$ be the fraction field of the henselization of $\mathbb{A}^1_{\kappa}$ at $s$. Let $l$ be a prime different from $\text{char}(\kappa)$. Assume that the inertia of $\text{Gal}(K)$ acts unipotently on $H^i(X_{\bar{K}},\mathbb{Q}_l)$.
Let $\tilde{\mathbb{A}}^1_{\mathbb{C}}$ be the universal covering space of $(\mathbb{A}^1_{\mathbb{C}})^{\text{an}}$, and let $\tilde{X}_{\mathbb{C}} = X_{\mathbb{C}} \times_{\mathbb{A}^1_{\mathbb{C}}} \tilde{\mathbb{A}}^1_{\mathbb{C}}$.
On one hand, there is a (Galois-theoretic) weight-monodromy filtration on $H^i(X_{\bar{K}},\mathbb{Q}_l)$.
On another hand, there is a (Hodge-theoretic) weight-monodromy filtration on $H^i(\tilde{X}_{\mathbb{C}},\mathbb{Q})$.
Question: Is it possible to (canonically) compare these analogous filtrations directly, e.g. using base change theorems?
(Also, corrections to any mistakes in the formulations above are welcome.)
