I do understand the fermentation might continue (subdued) at above fridge temp levels. That is fine...we have a sour tooth.As usual, it all depends. TONS of variables. I sense the direction that Holly suggests is about optimizing things for safety. IOW sure you could probably keep lacto fermented veggies over the fall and winter but longer than that will require stabilization (vinegar and canning). Refrigeration would also work to keep critters in check. Keep in mind the flavor is affected and most likely many would prefer the flavor of the stabilized stuff over au natural and I would do it for this reason alone.
I don't have that much to last till spring anyway.
My reluctance to caning or using vinegar comes from the desire to keep the probiotics alive. Fridge is an option too, but i was looking for a sustainable solution.