Conjecture C_st for analytic varieties

The relationship between p-adic (pro)-etale cohomology and de Rham cohomologies is well understood now for algebraic varieties over local fields of mixed characteristic. For analytic varieties it remains largely conjectural though much progress has been made in the last decade especially in the Stein case. I will focus on the etale-to-de Rham direction of the relationship for partially proper varieties where a progress was made recently via an introduction of a Galois  acyclic version of the period ring B_st. This is based on a joint work with Pierre Colmez,  Sally Gilles, and Zhenghui Li.