Local epsilon-isomorphism conjecture

Fix a prime $l$. For a coefficient ring $R$, an $R$-representation $T$ of $G_{\mathbb{Q}_l}$, and a $\mathbb{Z}_l$-basis $\zeta$ of $\Gamma(\overline{\mathbb{Q}}_p,\mathbb{Z}_l(1))$, let $\Delta_R(T)$ denote the associated determinant module. Local epsilon-isomorphism conjecture. There exists a unique compatible family

$$\varepsilon_{R,\zeta}(T):\bold{1}_R\overset{\sim}{\rightarrow}\Delta_R(T)$$

for all triples $(R,T,\zeta)$ satisfying the source's five properties: compatibility with base change, multiplicativity in exact sequences, the prescribed change under $\zeta\mapsto\zeta^a$, compatibility with duality, and agreement with the de Rham epsilon-isomorphism for de Rham representations. The conjecture is known when $l\ne p$ by Yasuda's construction; the $p$-adic case is the remaining part addressed in the paper.