The Fukaya–Kato local epsilon-isomorphism conjecture

Let $L/\mathbb Q_p$ be finite, let $V$ be a finite-dimensional $L$-representation of $G_{\mathbb Q_p}$, and let $T$ be a Galois-stable $\mathcal O_L$-lattice. Let $G$ be the relevant $p$-adic Lie group, $\Lambda=\Lambda(G)$, $\widetilde\Lambda=W(\overline{\mathbb F_p})[[G]]$, and $\mathbb T=\Lambda\otimes_{\mathbb Z_p}T$. Assume $T(\rho^*)$ is de Rham for the representations under consideration. Fukaya–Kato local epsilon-isomorphism conjecture. There exists a unique isomorphism in $\mathcal C_{\widetilde\Lambda}$

$$\epsilon_{p,\Lambda}(\mathbb T):\mathbf 1_{\widetilde\Lambda}\to\bigl(\mathbf d_\Lambda(\operatorname{R}\Gamma(\mathbb Q_p,\mathbb T))\mathbf d_\Lambda(\mathbb T)\bigr)_{\widetilde\Lambda}$$

such that for every $\rho:G\to\operatorname{GL}_n(\mathcal O)\subseteq\operatorname{GL}_n(L)$,

$$L^n\otimes_\Lambda\epsilon_{p,\Lambda}(\mathbb T)=\epsilon_{p,L}(T(\rho^*)).$$

This is a local integrality and interpolation conjecture for epsilon factors in non-commutative Iwasawa theory; it remains conjectural in the stated generality.