Variances of child time series do not sum to variance of parent time series

I'm attempting a version of [top-down reconciliation](https://otexts.com/fpp3/single-level.html). I have a root-level time series with all of my data, and then the next level is product category (I'm working with demand data). The theory is that the sum of the distributions of the child models (product category) should sum to the parent model (total demand). The reconciliation process works just fine with yhat--only small adjustments are made.

But I also need information on the variance and although in theory, variances of normal distributions should sum to the parent distribution, this does not seem to be the case here. I'm finding the variance by setting the prediction interval to 0.6827 (one-sigma), subtracting `yhat_lower` from `yhat_upper`, dividing by two, and then squaring. I've even pulled the predictive samples and calculated the variance directly to find roughly the same value. But the sum of the variances of the child-level models is much lower than the variance of the parent model.

Is Prophet performing some kind of extra manipulation that I'm not taking into account that would result in this discrepancy? Or have I fundamentally misunderstood normal distributions?