Proof of GMM System and Difference selection of high values in misspecified AR(1) models

Here's the proof that the GMM System and GMM Difference estimators select high values from the range of an autoregressive parameter in an AR(1) model, where the autoregressive parameter is assumed to be constant across groups but isn't. There's an assumption of ergodicity. They are close to a general proof for all GMM estimators with parameter-linear orthogonality conditions, and the proof that the GMM System selects higher values than the Difference estimator should follow from the result that (a/b > c/d implies a/b > (a+c)/(b+d)). But I haven't pushed these results through.

The results are in image form because there are some formulae which don't paste directly.