Inspired by Nakamura’s work 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 $D_{\text{cris}}$ and $D_{dR}$, respectively. For rank one modules of the above type or slightly more general for trianguline ones, we construct $\epsilon$-isomorphisms for their Lubin-Tate deformations satisfying the desired interpolation property. This is joint work with Milan Malcic, Rustam Steingart and Max Witzelsperger.