An equivariant Iwasawa main conjecture for local fields

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.