There is the Chromatic Convergence Theorem by Hopkins and Ravanel which states that the homotopy inverse limit of the chromatic tower of a finite spectra $X$ is $X$.
Let's call any spectra with this property chromatically complete
There is an article from T. Barthel (Chromatic Completion) which proves a slightly stronger result: to make the tower converge, the spectra $X$ does not have to be finite, but has to have finite projective $BP$-dimension.
The same article also concludes saying that they do not know whether all suspension spectra are chromatically complete and that they believe that the collection of chromatically complete spectra is not closed under infinite products.
So here are my two questions:
Since this article is more than two years old, have we made some progresses about proving or not those two properties?
Besides finite spectra and spectra with finite projective $BP$-dimension, do we know other kinds of spectra that are chromatically complete?