A liqueur sweetened with sugar cane, its flavour comes from an infusion of botanicals including lime peel, almonds and cloves. Developed by Sir John Taylor...
It is only 11% alc./vo./, so depending on storage conditions, it does not have enough alcohol to prevent mould forming. So yes, I'd refrigerate. OR use a more alcoholic Falermum like our own Difford's Falernum, which is 18% alc./vol. No mould will form on a product over 17% alc./vol. That's why Bailey's liqueur is 17%!
If you can't buy Difford's Falernum, you could bring up the proof of Taylor's to 18%. Allowing for contraction and staying at around 700ml total, using 96% alcohol, you'd need to remove 58ml of Falernum (and make something nice), add in 58ml of 96% and you'd have 693.7ml of 18%.
Or use (more) OP rum: e.g. Wray and Nephew OP at 63% would need 95ml or Plantation Original Dark at 73% would need 80ml.
IDK the taste profiles though of the new versions, you could do a smaller sample to experiment.