This is something which I'm sure is well known to experts which I would appreciate some information about. In his paper [1], Scholze proves (e.g. Theorem 8.4, Theorem 8.8) that on a proper adic space $X$ over $\textrm{Spa}(k,\mathcal{O}_{k})$, with $k$ an appropriate non-archimedean field, the natural map $H^{i}(X_{\textrm{pet}}, \hat{\mathbb{Z}}_{p}) \otimes B_{\textrm{dR}} \to H^{i}(X_{\textrm{pet}}, \mathbb{B}^{+}_{\textrm{dR}})$ is an isomorphism, where I use the notation "pet" for the pro-etale site. I think a similar statement should be true for $X$ the adic space associated to a smooth algebraic variety over $k$, but I cannot find the appropriate reference.
I am aware of work [2] of Diao, Lan, Liu and Zhu which works with a general smooth variety, but I am really interested in the "intermediate" comparison between cohomology wich coefficients in $\hat{\mathbb{Z}}_{p}$ and $\mathbb{B}^{+}_{\textrm{dR}}$ rather than the actual comparison isomorphism of $p$-adic Hodge theory, and it seems to me they only handle this intermediate fact in the proper case. The related work [3] seems to do something like what I want in Proposition 3.3.4, but in a compactly supported setting which isn't really what I'm interested in.
If it helps I am really just looking to understand the maps $H^{i}(X_{\textrm{pet}}, \hat{\mathbb{Z}}_{p}) \otimes B_{\textrm{dR}} \to H^{i}(X_{\textrm{pet}}, \mathbb{B}^{+}_{\textrm{dR}})$ when $X = \mathbb{G}_{m}^{w}$ is the $w$-fold product of the multiplicative group, so an explicit description of this map would also be appreciated.
Edit: I may have found what I want in Theorem 7.14 of [4]; I will take a look at the paper and report back.