Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy filtration on the $i$th cohomology groups $H^i_\mathrm{et}(X_{\overline{\mathbb{Q}}_p},\mathbb{Q}_\ell)$ are pure of weight $i+k$.

I was wondering if there is a similar conjecture for open varieties, coming from looking at a good compactification $X\hookrightarrow \overline{X}$. If $X$ is a curve, then I can get my hands on what's happening explicitly, so suppose that $X$ is a curve with compactification $\overline{X}$ and $D$ is the set of missing points.

Then the only interesting group is $H^1_\mathrm{et}(X_{\overline{\mathbb{Q}}_p},\mathbb{Q}_\ell)$ which by excision sits in an exact sequence $$ 0 \rightarrow H^1_\mathrm{et}(\overline{X}_{\overline{\mathbb{Q}}_p},\mathbb{Q}_\ell) \rightarrow H^1_\mathrm{et}(X_{\overline{\mathbb{Q}}_p},\mathbb{Q}_\ell) \rightarrow V \rightarrow 0 $$ where $V$ is a subspace of $H^0_\mathrm{et}(D_{\overline{\mathbb{Q}}_p},\mathbb{Q}_\ell)(-1)$, it is therefore pure of weight $2$. So (pretty tautologically) $V$ turns up in the 2nd graded piece $\mathrm{Gr}_2^W$ of the weight filtration, but with some basic linear algebra I think I've convinced myself that $V$ actually turns up in the 0th graded piece $\mathrm{Gr}_0^M$ of the monodromy filtration on $H^1_\mathrm{et}(X_{\overline{\mathbb{Q}}_p},\mathbb{Q}_\ell)$, and apart from this 'extra' $V$, the graded pieces of the monodromy filtration for $X$ are the same as those for $\overline{X}$.

In other words, $\mathrm{Gr}^M_{-1}$ and $\mathrm{Gr}_1^M$ are pure of weights 0 and 2 respectively, but $\mathrm{Gr}^M_1$ is mixed with weights 1,2. So the naive weight monodromy conjecture fails, but you can still say something - the graded pieces $\mathrm{Gr}_k^M$ are mixed of weights $\geq i+k$.

So I guess this is my question: is there a conjectural 'weight monodromy' for open varieties which says something like the above, e.g. the $k$th graded pieces is mixed with weights in $[i+k,2i+k]$? Maybe this would easily follow from the 'usual' weight monodromy by choosing a good compactification and then just using some linear algebra?