$\epsilon$-isomorphisms for rank one $(\varphi,\Gamma)$-modules over Lubin-Tate Robba rings

Inspired by Nakamura's work (arXiv:1305.0880) on $\epsilon$-isomorphisms for $(\varphi,\Gamma)$-modules over (relative) Robba rings with respect to the cyclotomic theory, we formulate an analogous conjecture for $L$-analytic Lubin-Tate $(\varphi_L,\Gamma_L)$-modules over (relative) Robba rings for any finite extension $L$ of $\mathbb{Q}_p.$ In contrast to Kato's and Nakamura's setting, our conjecture involves $L$-analytic cohomology instead of continuous cohomology within the generalized Herr complex. Similarly, we restrict to the identity components of $D_{cris}$ and $D_{dR},$ respectively. For rank one modules of the above type or slightly more generally for trianguline ones, we construct $\epsilon$-isomorphisms for their Lubin-Tate deformations satisfying the desired interpolation property.