one thing that you need to understand, and not underestimate, is that when saving to a different file format, transformation occurs meaning that the color information can be part or not of the file saving option. You also need to keep in mind that file reading depends on the software vender too.
ie there are 3 stages in data translation:
each stage introduces instability and depends on the software vendor and the file format definition.
With the above this is no wonder that the result may vary.
The best practice, as far as my experience can tell, is to try all options.