Suppose $T$ is a free finite rank $\mathbb{Z}_p$-module with a continuous action of $\operatorname{Gal}(\overline{K} / K)$, where $K$ is a number field. There is a definition of local Tamagawa numbers $\operatorname{Tam}(T / K_v)$ for each prime $v$ of $K$, going back to Fontaine and Perrin-Riou (or to Bloch and Kato for $K = \mathbb{Q}$). For $v \nmid p$ this is the order of the torsion subgroup of $H^1(I_v, T)^{D_v}$, where $D_v$ and $I_v$ are the decomposition group and intertia group at $v$; for $v \mid p$ it is something more complicated using the Bloch-Kato exponential map. (I'm led to believe that if $T$ is the Tate module of an abelian variety, this recovers the usual description in terms of Neron models.)

If $K_{v, n} = K_v(\mu_{p^n})$, is it true that the factors $\operatorname{Tam}(T / K_{v, n})$ are eventually constant for large enough $n$?

(EDIT: In the light of Rob's comment, maybe I should add the assumption that my Galois representation is crystalline at $p$. I'm chiefly interested in the case of the p-adic representation of a modular forms of level prime to $p$ and non-ordinary at $p$.)