Locally analytic period rings as power-series rings in the cyclotomic case

Let $K$ be a $p$-adic field, let $K_\infty/K$ be an infinitely ramified $p$-adic Lie extension containing a cyclotomic extension, and let $d$ be the dimension of its Galois group $\Gamma_K$ as a $p$-adic Lie group. For an interval $I$, write $({\tilde{\mathbf{B}}}_K^I)^{{\mathrm{la}}}$ for the locally analytic vectors in the corresponding higher period ring. Locally analytic period-ring conjecture. The rings $({\tilde{\mathbf{B}}}_K^I)^{{\mathrm{la}}}$ are the completion for the locally analytic topology of rings of power series in $d$ variables. This is the expected structure needed to generalize $(\varphi,\Gamma_K)$-module theory to infinitely ramified $p$-adic Lie extensions; the source presents it as an expectation rather than establishing it.