If $K_n$ is the field $\mathbb{Q}_p(\mu_{p^n})$, then it's easy to see that the relative different $\mathcal{D}(K_n / K_{n-1})$ is $(p)$ for all $n \ge 2$.
What happens if I take an arbitrary, probably totally ramified, field finite extension $L$ L/\mathbb{Q}_p$ and look at the tower $L_n = LK_n$? It's clear that $\mathcal{D}(L_n / L_{n-1})$ divides $(p)$, and one can show (using a general result of Tate on $\mathbb{Z}_p$-extensions) that its valuation tends to 1 as $n \to \infty$; but is it true that it's equal to $(p)$ for all sufficiently large $n$?
