Yea the house (aux) batts should be charged with the engine running.
It is controlled by a BCC (battery control center). There are conditions both batts are supposed to link and charge each other. Whether they are or not is the question. There are electronics involved that control this and electronics fail. Also why having a multimeter can help figure out where the problem is. If you search the forum for BCC youíll see lots of info breads on the topic.
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