Before revision:  Our bound even holds under the Pair Conditional Sampling model, a restricted version of the conditional sampling model where every queried subset S either must have exactly 2 elements, or must be the entire domain of the distribution. We also prove that tolerant identity testing in the conditional sampling model can be solved in \tilde{O}(\varepsilon^{-4}) queries, which is the first known bound independent of the support size of the distribution for this problem.
After revision:  Our bound even holds under the Pair Conditional Sampling model, a restricted version of the conditional sampling model called the Pair Conditional Sampling model. Next, we prove that tolerant identity testing in the conditional sampling model can be solved in \tilde{O}(\varepsilon^{-4}) queries, which is the first known bound independent of the support size of the distribution for this problem.