  Let $L/K$ be a finite Galois extension of $p$-adic fields and let
$L_{\infty}$ be the unramified $\mathbb Z_p$-extension of $L$. Then
$L_{\infty}/K$ is a one-dimensional $p$-adic Lie extension. In the spirit of
the main conjectures of equivariant Iwasawa theory, we formulate a conjecture
which relates the equivariant local epsilon constants attached to the finite
Galois intermediate extensions $M/K$ of $L_{\infty}/K$ to a natural arithmetic
invariant arising from the \'etale cohomology of the constant sheaf $\mathbb
Q_p/\mathbb Z_p$ on the spectrum of $L_{\infty}$. We give strong evidence of
the conjecture including a full proof in the case that $L/K$ is at most tamely
ramified.
