Should candy cane-infused rum be kept in the fridge?

I’m looking to make a fun cocktail for my friends next weekend, and I thought a good start would be to infuse some black rum (47%/94 proof) with some crushed up candy canes.

I washed out an old jar (NOT airtight, but definitely water tight), threw in a couple shots of rum and some crushed up candy canes. They dissolved much faster than I thought, which got me thinking, should this go in the fridge?

I was assuming that the alcohol content would remove risk for mold, but given the jar isn’t airtight should I play it safe? Never infused before, sorry if this is a noob question