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

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 $n\geq1$, $V=K(\chi_{\pi})$, $h_{F_n,K(\chi_{\pi})}^1{\partial {F(T)}u})=(q/\pi)^{-n}\delta(x_n)$ and any $j\leq{-1}$, $$\exp^*_{F_n，V^*(1-j)}(\partial {F(T)}u\otimes{e_j}))=\frac{1}{(-1-j)!}(\pi/q)^n\partial_{V(\chi_\pi^j)}(\partial^{-j}f(T)\otimes{t_\pi^{-j-1}e_{1+j}}).$$ It seems that the LHS is independent of the choice of $f(T)$, while the RHS determines $\varphi^{-q}(f(T))\in{K_n[[t_{\pi}]]}$ up to constants, what's wrong? Thanks. All my bests.