As I indicated in this post, lots of good information is available in "unpublished" NASA memorandum.
So Chao essentially reports that the Chandler wobble is in-phase with ENSO from 1900-1979, but it flips phase starting around 1980. This is precisely in line with what I am finding.
Here's an example of how you can gain confidence that you are on the right track. I configured the multiple linear regression to train on the ENSO time series applying the wave equation transform from 1880-1940, and then looked at the correlation coefficient over the interval 1940-2013 -- which is completely outside the training interval.
The in=inside cc is 0.725 and the out=outside cc is 0.758. It is rare that you can get a correlation higher outside of the training interval, but because of the stronger noise in the earliest ENSO measurements this is not out of the question. In terms of variance contributions, the reduced noise promotes the real underlying signal.
Next, we generate a set of several synthesized red-noise random walks that have similar variance to the actual ENSO data. This is essentially the Ornstein-Uhlenbeck algorithm, describing a random walk with a reversion-to-the-mean potential well factor. Ten runs were combined into an animated GIF shown below:
Note that although the correlation coefficient is moderate within the training interval (anywhere from 0.4 to 0.6), it is insignificant outside of the training interval, and a few times it actually goes negative. In other words, there is no phase coherence outside of the interval, and any apparent agreement within the training interval is likely the result of over-fitting.
Alas, research is still being published  arguing whether or not ENSO is red noise. This is actually a moot point since what the GCMs generate (as far as I can tell) are invariably reported as sets of time series with random outcomes. If ENSO is actually deterministic -- apart from the rare metastable phase reversal and volcanic activity -- the GCMs should be slight variations of Figure 2 and not Figure 3.
 Chen, Xianyao, and John M. Wallace. "Orthogonal PDO and ENSO indexes." Journal of Climate 2016 (2016).
You don't need a weatherman to know which way the wind blows.