Consequence of $\mathsf{NP\subseteq BPP}$ to $\mathsf{NP\subseteq ZPP}$?
If you could prove that implication, then lower bounds such as $NEXP \neq BPP$ would hold unconditionally. (proof: $NEXP = BPP$ implies $NEXP = \Sigma_2 P$ and $NP \subset BPP$, and if the latter also implies $NP$ is in $coRP$ then UNSAT is in $NP$. So $NEXP = \Sigma_2 P = NP$ which is false by the nondeterministic time hierarchy). Good luck!