This is a followup to this question.

Let $p \ge 3$ be prime, and let $V$ be a crystalline 2-dimensional representation of $G_{\mathbb{Q}_p}$ and $T$ a lattice in $V$. I'm going to assume just about every niceness condition on $V$ that I can think of:

$V$ is irreducible;

$\operatorname{Fil}^0 \mathbb{D}_{\mathrm{cris}}(V)$ is 1-dimensional (so one Hodge-Tate weight of $V$ is $\le 0$ and the other is $> 0$)

none of the eigenvalues of Frobenius on $\mathbb{D}_\mathrm{cris}(V)$ are integral powers of $p$;

the Hodge filtration of $V$ has length $< (p-1)$, so $T$ corresponds to a strongly divisible $\mathbb{Z}_p$-lattice $\mathbb{D}(T)$ in $\mathbb{D}_{\mathrm{cris}}(V)$ via Fontaine-Laffaille theory.

Let $T$ be a lattice in $V$, and let $\omega$ be a $\mathbb{Z}_p$-basis of the "tangent space" $t_T = \mathbb{D}(T) / \operatorname{Fil}^0 \mathbb{D}(T)$. The Tamagawa number of $T$ over $K_n = \mathbb{Q}_p(\mu_{p^n})$ is given by $$ \operatorname{Tam}^0_{K_n, \omega}(T) = \frac{[H^1_f(K_n, T) : \exp(\mathcal{O}_{K_n} \omega)]}{[\mathbb{D}(T) : (1- \varphi) \mathbb{D}(T)]}$$

where $[ A : B ]$ is a generalised index (so $[ \mathbb{Z}_p : \tfrac{1}{p} \mathbb{Z}_p] = \tfrac{1}{p}$ etc).

Question: Is it true that under the above hypotheses this Tamagawa number is always 1?

I know this is true for all $n$ if $V$ corresponds to an elliptic curve (because the Tamagawa number has an alternative definition in terms of the index of the nonsingular points in the special fibre of the Neron model) and, if I've understood correctly, for $n = 0$ it is true for any $V$ satisfying the hypotheses above (by a theorem of Bloch and Kato).