On the cohomology of p-adic analytic spaces, II: The $C_{\rm st}$-conjecture

Long ago, Fontaine formulated conjectures (now theorems) relating \'etale and de Rham cohomologies of algebraic varieties over $p$-adic fields. In an earlier work we have shown that pro-\'etale and de Rham cohomologies of analytic varieties in the two extreme cases: proper and Stein, are also related. In the proper case, the comparison theorems are similar to those for algebraic varieties, but for Stein varieties they are quite different. In this paper, we state analogs of Fontaine's conjectures for general smooth dagger varieties, that interpolate between these two extreme cases, and we prove them for many ``small'' varieties (cases we deal with include products of overconvergent affinoids and proper varieties, analytifications of algebraic varieties, or "almost proper" dagger varieties). The proofs use a ``geometrization'' of all involved cohomologies in terms of quasi-Banach-Colmez spaces (qBC's for short, quasi- because we relax the finiteness conditions). The heart of the proof relies on delicate properties of BC's and qBC's. These properties should be of independent interest and we have devoted a large part of the paper to them.