PaulP:"You think I care one iota about being able to predict the next few ENSO cycles? Taking that by itself, predicting the next few cycles will only raise the question as whether that happened purely by chance."
You got me there; thought you cared. Would you not rightly celebrate if your Model duly predicted upcoming cycles? You instead raise the question whether your Model is truly predictive, by willfully not predicting any cycle right as it comes.
I question whether any correct Model would likely long fail to predict cycles, "purely by chance", and predict the Lunisolar signal will be mostly seen as sensor noise in ENSO data, past, present, and future. Let the question therefore be, "whether that happened purely by chance". Not likely.
As for identifying the common-mode of ENSO, QBO, Chandler Wobble, MJO, TIW, AMO, PDO, PNA, NAO, IOD, AO, AAO; these are all spring-mass oscillators, either singly or as networks; and Yes, the Lunisolar signal would appear as noise in all their sensor data networks, as well as adding a some excitation, and even a tiny bit of period-forcing in the multi-chaos statistics.