Let $V$ be a continuous representation of the absolute Galois group of $\mathbb{Q}_p$ with coefficients in $\mathbb{Q}_p$. The theory of Sen attaches to $V$ generalized Hodge-Tate weights which are elements in $\overline{\mathbb{Q}}_p$.

**My question is**: are the generalized Hodge-Tate weight always elements of $\overline{\mathbb{Z}}_p$?

In his infinite fern paper from 1997, Mazur says he does not know of an example (at least in dimension 2) where the weights are not in $\overline{\mathbb{Z}}_p$. Personally, I do not recall of ever seeing an example with weights not lying in $\mathbb{Z}_p$.

Thanks!