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.
If you take advantage of our quantity discounts and store many jars you might consider storing the unopened jars in a cool dry place, refrigerator or freezer. The cooler the better for long term storage.
The same philosophy applies to the butter oil products.