  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.
