The étale cohomology $R\Gamma_{\mathrm{ét}}(X;\mathbb{Z}_p(n))$ of a scheme $X/K$ can be computed by a Hochschild-Serre spectral sequence with terms of the form $H^i(K;H^j(X_{\overline{K}};\mathbb{Z}_p)(n))$.
For $K$ a number field and $2n-i \neq 0$, the map $R\Gamma_{\mathrm{mot}}(X;\mathbb{Z}(n)) \otimes \mathbb{Z}_p \to R\Gamma_{\mathrm{ét}}(X;\mathbb{Z}_p(n))$ induces maps $H^i_{\mathrm{mot}}(X;\mathbb{Z}(n)) \otimes \mathbb{Z}_p \to H^1(K;H^{i-1}(X_{\overline{K}};\mathbb{Z}_p)(n))$, but this latter map is rarely an isomorphism, even rationally, since its image lies in the subgroup $H^1_g(K;H^{i-1}(X_{\overline{K}};\mathbb{Z}_p)(n))$.
Has anyone ever discussed a Bloch-Kato version of $R\Gamma_{\mathrm{ét}}(X;\mathbb{Z}_p(n))$? That is, with a spectral sequence computed by $H^i_g(K;H^j(X_{\overline{K}};\mathbb{Z}_p)(n))$.
For simplicity, let's suppose $X$ is smooth over $K=\mathbb{Q}$ and work rationally. The most natural in my mind would be to let $S$ be an arbitrary finite set of primes containing $p$ and outside of which $X$ has a good reduction model $\mathscr{X}_S$, then define $R\Gamma_{\mathrm{f,S}}(X;\mathbb{Q}_p(n))$ to be the pullback of the natural maps $R\Gamma_{\mathrm{ét}}(\mathscr{X}_S;\mathbb{Q}_p(n)) \to R\Gamma_{\mathrm{ét}}(X_{\mathbb{Q}_p};\mathbb{Q}_p(n))$ and $R\Gamma_{\mathrm{syn}}((X_{\mathbb{Q}_p})_h,n) \to R\Gamma_{\mathrm{ét}}(X_{\mathbb{Q}_p};\mathbb{Q}_p(n))$, the latter of which appears in Theorem A of https://arxiv.org/pdf/1309.7620.pdf. Then $R\Gamma_{\mathrm{g}}(X;\mathbb{Q}_p(n))$ should be the direct limit of $R\Gamma_{\mathrm{f,S}}(X;\mathbb{Q}_p(n))$ as $S$ increases.
With this definition, it is clear (from the paper mentioned) that $R\Gamma_{\mathrm{mot}}(X;\mathbb{Z}(n)) \otimes \mathbb{Q}_p$ maps to $R\Gamma_{\mathrm{g}}(X;\mathbb{Q}_p(n))$. A natural generalization (rephrasing?) of the conjecture of Bloch-Kato (Conjecture 5.3 of https://virtualmath1.stanford.edu/~conrad/BSDseminar/refs/BKTamagawa.pdf) would be that this map is an isomorphism. Has this idea ever been written out anywhere?
Another natural consideration is: does this construction see the exceptional zeroes of $p$-adic L-functions the way that Nekovar's Selmer Complexes do?
