Clausen's conjecture on analytic syntomic cohomology of the $p$-adic complex numbers

For an integer $n\geq 2$, let $R\Gamma(\mathbb{C}_p^\mathrm{Syn},\mathcal{O}\{n\})$ denote analytic syntomic cohomology of $\mathbb{C}_p$ with coefficients in the $n$-th Tate-twisted structure sheaf, and let $B_\mathrm{crys}^+$ be the crystalline period ring with Frobenius $\phi$. The notation $\mathbb{Z}_p(n)[-1]$ denotes the $n$-th Tate twist shifted by $-1$. Clausen's conjecture. For $n\geq 2$, one has

$$R\Gamma(\mathbb{C}_p^\mathrm{Syn},\mathcal{O}\{n\})\cong (B_\mathrm{crys}^+)^{\phi=p^n}/\mathbb{Z}_p(n)[-1].$$

This conjecture is motivated by calculations of the $K$-theory of nuclear solid modules over $\mathbb{C}_p$ and an expected Atiyah--Hirzebruch spectral sequence relating that analytic $K$-theory to analytic syntomic cohomology. Its status is not determined in the supplied source.