Nakayama's local epsilon conjecture for de Rham $(\varphi,\Gamma)$-modules

Let $L$ be a finite extension of $\boldsymbol{Q}_p$, let $D$ be a de Rham $(\varphi,\Gamma)$-module over the Robba ring $\mathcal{R}_L$, and let $\Delta_L^{\operatorname{Iw}}(D)$ be its cyclotomic deformation fundamental line. For every de Rham continuous character $\delta:\Gamma\to L^\times$, let $D(\delta)$ denote the twist and let $\varepsilon_L^{\operatorname{dR}}(D(\delta)):1_L\xrightarrow{\sim}\Delta_L(D(\delta))$ be the de Rham epsilon-isomorphism. The specialization map is denoted by $\operatorname{ev}_\delta$.

Nakayama's local epsilon conjecture. There exists an isomorphism

$$\varepsilon_L^{\operatorname{Iw}}(D):1_{\mathcal{R}_L^+(\Gamma)}\xrightarrow{\sim}\Delta_L^{\operatorname{Iw}}(D)$$

such that, for every de Rham continuous character $\delta$, the diagram comparing specialization along $f_\delta$ with $\varepsilon_L^{\operatorname{dR}}(D(\delta))$ commutes:

$$\begin{CD} \Delta_L^{\operatorname{Iw}}(D)\otimes_{f_\delta}L @>{\operatorname{ev}_\delta}>> \Delta_L(D(\delta))\\ @V{\varepsilon_L^{\operatorname{Iw}}(D)\otimes\operatorname{id}}VV @VV{\varepsilon_L^{\operatorname{dR}}(D(\delta))}V\\ 1_{\mathcal{R}_L^+(\Gamma)}\otimes_{f_\delta}L @>{\operatorname{can}}>>1_L \end{CD}$$

This is the explicit cyclotomic-deformation formulation of the local epsilon conjecture. It asserts compatibility of a universal epsilon-isomorphism with all de Rham specializations; the supplied text does not state whether this formulation has been resolved.