The Efron-Stein inequality for identically distributed pairs

We prove that the classical Efron--Stein inequality holds for independent exchangeable pairs $(X_i,Y_i)$. The same inequality fails for independent identically distributed pairs; a simple trigonometric counterexample shows that the trivial Cauchy--Schwarz bound of factor $n$ is sharp. When each random variable takes at most $k_i$ values, a useful bound still holds with explicit constant $\rho(k)\le\max_i k_i/2$.