| Abstract: Inspired by Nakamura's work (arXiv:1305.0880) on epsilon-isomorphisms for (ɸ ,ℾ)-modules over (relative) Robba rings with respect to the cyclotomic theory, we formulate an analogous conjecture for L-analytic Lubin-Tate (ɸ ,ℾ)-modules over (relative) Robba rings for any finite extension L of Qp. 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 Dcris and DdR, 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 |