Proof.³³ 3 We thank to Ofer Gabber for suggesting the proof below It suffices to prove that $n$-dimensional cells are dense in $B$, where $n=\dim X$. For any open $U\subset B,\,\,U\neq\emptyset$ we have $H_{c}^{n}(U,\pi_{\ast}(\Omega^{n}_{X}))\simeq H_{c}^{n}(\pi^{-1}(U),\Omega^{n}_{X})$.

The last group is nontrivial, because for any non-empty open $V\subset X^{an}$ the integration map $\int:H_{c}^{n}(V,\Omega^{n}_{X})\to K$ is onto. Therefore $\dim(U)\geq n$. $\blacksquare$