$B_{\mathrm{cris}}\subseteq B_{\mathrm{dR}}$ and $B_{\mathrm{dR}}^+$ are well-known period rings in $p$-adic Hodge. I know $B_{\mathrm{dR}}=B_{\mathrm{dR}}^+[\frac{1}{t}]$ and $\frac{1}{t}\in B_{\mathrm{cris}}$ where $t$ is Fontaine's $2\pi i$.

I want to ask if the natural map $B_{\mathrm{cris}}\rightarrow \frac{B_{\mathrm{dR}}}{B_{\mathrm{dR}}^+}$ induced by the inclusion is surjective. This is to say if $B_{\mathrm{dR}}=B_{\mathrm{cris}}+{B_{\mathrm{dR}}^+}$. For example, if $x\in B_{\mathrm{dR}}^+$, then $\frac{x}{t}\in B_{\mathrm{cris}}+{B_{\mathrm{dR}}^+}$?

Thanks!