In the paper [Iwasawa Theory and F-analytic Lubin-tate $(\phi,\Gamma)$-modules
][1]

Prop 3.4.2 says that for any $x\in{S}$, there exists (not uniuqely) $f(T)\in{B_{rig,F}^+}$ such that
$f(u_n)=\log_{LT}(x_n)$ and $\psi_{q}(f(T))=\pi^{-1}{f(T)}$. Thm 3.4.5 and Thm 3.5.3 says that for any $n\geq1$, $V=K(\chi_{\pi})$, take $y=f(T)\otimes{t_\pi^{-1}u}$, we have
$h_{F_n,V}^1(\partial{f(T)}u)=(q/\pi)^{-n}\delta(x_n)$ and any $j\leq{-1}$,
$$\exp^*_{F_n，V^*(1-j)}(h_{F_n,V(\chi_{\pi}^j)}(\partial {f(T)}u\otimes{e_j}))=\frac{1}{(-1-j)!}(\pi/q)^n\partial_{V(\chi_\pi^j)}(\partial^{-j}f(T)\otimes{t_\pi^{-j-1}e_{1+j}}).$$
It seems that the LHS is independent of the choice of $f(T)$ (it is the twist (a la soule) of the system $(q/\pi)^{-n}\delta(x_n)$ by $\chi_{\pi}^j$),  while the RHS determines $\varphi^{-q}(f(T))\in{K_n[[t_{\pi}]]}$ up to constants, what's wrong? Thanks.
All my bests.


  [1]: http://perso.ens-lyon.fr/laurent.berger/articles/article29.pdf