Search versus Decision for $\mathsf{S}_2^\mathsf{P}$

We compare the complexity of the search and decision problems for the complexity class S2P. While Cai (2007) showed that the decision problem is contained in ZPP^NP, we show that the search problem is equivalent to TFNP^NP, the class of total search problems verifiable in polynomial time with an NP oracle. This highlights a significant contrast: if search reduces to decision for S2P, then $Σ_2^p \cap Π_2^p$ is contained in ZPP^NP.