Another question; are prediction intervals in bottom-up made at the total level, or just at the bottom level and then summed up? If so, how does the summing up of the prediction intervals work

How prediction intervals are constructed depends on which method is chosen for creating prediction intervals. There's three methods: Normality, Bootstrap and PERMBU. You can find how they work here, there's also a link to the source paper of each of these methods in there. The default is Normality. This assumes all timeseries follow a Gaussian distribution. It samples from a multivariate normal for each series. The intervals are subsequently constructed by quantiling the samples. So, the intervals are made at every level by generating coherent samples for each level.

In this case its conformal prediction. I can not find anything on that method. How would it work in that case?

It's not conformal prediction. You're asking about prediction intervals in hierarchical forecasting. We don't support conformal there.

https://nixtlaverse.nixtla.io/hierarchicalforecast/examples/mlframeworksexample.html are you sure? Because I do see conformal prediction

That example makes prediction intervals with MLForecast. They are then reconciled with hierarchicalforecast using the methods I described above.

That is incorrect

Curious to understand what is incorrect