The [SAM](https://en.wikipedia.org/wiki/Antarctic_oscillation) index ([data source](https://climatedataguide.ucar.edu/climate-data/marshall-southern-annular-mode-sam-index-station-based)) is the last to do on the map
Based of the complexity of these waveforms, this should have taken a long time to adequately fit a model if starting from scratch. Yet, since the tidal forcing is nearly identical for each, the computation took no time at all.
The LTE modulation was close to that of the complementary AO, as it retains the same phase over a greater range of forcing levels (indicated by the yellow arrow):
The fit is very good (Fourier spectrum comparison in lower panel)
As a bottom-line, these climate indices are likely not related as teleconnections (which is the current consensus idea), but more likely by a common-mode forcing . The set is synchronized by the common lunisolar tidal forces operating across the earth and individually distinguished by the standing wave constraints of each region.
Moreover, it's highly unlikely that the quality of these model fits is due to overfitting as there are very few DOF available given the common-mode forcing constraint shared by each model.