  Let $N/K$ be a finite Galois extension of $p$-adic number fields and let
$\rho^\mathrm{nr} : G_K \to \mathrm{Gl}_r(\mathbb Z_p)$ be an $r$-dimensional
unramified representation of the absolute Galois group $G_K$ which is the
restriction of an unramified representation $\rho^\mathrm{nr}_{\mathbb Q_p} :
G_{\mathbb Q_p} \to \mathrm{Gl}_r(\mathbb Z_p)$. In this paper we consider the
$\mathrm{Gal}(N/K)$-equivariant local $\epsilon$-conjecture for the $p$-adic
representation $T = \mathbb Z_p^r(1)(\rho^\mathrm{nr})$. For example, if $A$ is
an abelian variety of dimension $r$ defined over $\mathbb Q_p$ with good
ordinary reduction, then the Tate module $T = T_p\hat A$ associated to the
formal group $\hat A$ of $A$ is a $p$-adic representation of this form. We
prove the conjecture for all tame extensions $N/K$ and a certain family of
weakly and wildly ramified extensions $N/K$. This generalizes previous work of
Izychev and Venjakob in the tame case and of the authors in the weakly and
wildly ramified case.
