The agreement property for independently trained classifiers

Let $\mathcal{F}$ be a classifier family and let $\mathcal{D}$ be a distribution. For independent train sets $S_1,S_2\sim\mathcal{D}^n$, let $f_1,f_2$ be classifiers trained on $S_1,S_2$, respectively. The test accuracy is the probability that $f_1(x)=y$, while the agreement property. For certain classifier families $\mathcal{F}$ and distributions $\mathcal{D}$, these probabilities are approximately equal:

$$\Pr[f_1(x)=y]\approx\Pr[f_1(x)=f_2(x)].$$

Moreover, with high probability over the trained classifiers $f_1,f_2$, the corresponding probabilities over $(x,y)\sim\mathcal{D}$ are approximately equal. This is proposed as an instantiation of the paper's indistinguishability conjecture; it is provable for 1-nearest neighbors in some settings, but the general claim's resolution is not specified.