Many call on this topic.
Fermented clo products: The oil is very stable. In my opinion many can store OPENED jars either way. If you use the fclo regularly (within a few months) you can opt to leave in a cupboard after opening.
We do not overly process the oils so some lots tend to become thicker as the storage temps get colder. I have seen some really cold refrigerators make a fclo almost Gel.