I saw it already, however I did not understand why to HP-filter the log-linearized model simulations, since they are log-deviations from the steady state values (which are all zero in my model).

Perhaps it is just for log-linearized models that have non-zero steady states… what do you think?

I will wait for Professor Pfeifer answer to be sure as well.

It’s not about the mean, but the frequency components. If you consider your model to be the data generating process that created the empirical data, then a meaningful comparison involves using the same filter for both the model and the data.
But there is an alternative approach that wants the DGSE model to just replicate the filtered data.
See e.g.