Definite answer (for ketchup, which is comparable):

Heinz says put it in the fridge for optimal results:

https://www.allrecipes.com/heinz-settles-ketchup-storage-debate-7556212

"There was only ever one correct answer, and we're happy to share with Heinz Tomato Ketchup lovers that our ketchup has to be in the fridge," Olivia Lennon, at Heinz Tomato, told Daily Mail.

"Although we're aware many Heinz Tomato Ketchup fans have been storing their ketchup in the cupboard, we do recommend refrigeration after opening," she clarified. "This is the best way to maintain the delicious tangy taste of our Heinz Tomato Ketchup that you know and love."

Heinz bottle labels in the U.S. clearly state, "For Best Results, Refrigerate After Opening."

The company addressed the issue in a similar poll on Twitter in 2017 about ketchup storage.

"Because of its natural acidity, Heinz Ketchup is shelf-stable," the brand said. This explains why the product is not found in the refrigerated section of your grocery store. However, the brand maintained that consumers should "refrigerate after opening to maintain product quality."

Perhaps less cautious non-corporate answer, depends on how fast you're using it:

https://cooking.nytimes.com/article/does-ketchup-need-to-be-refrigerated

"If you use it often and it’s shelf stable, keep it in the pantry or on the counter,"

Another answer:

https://www.bbcgoodfoodme.com/how-to/should-ketchup-be-refrigerated/

"Opened Bottles: Once a bottle of ketchup is opened, the dynamics change. Exposure to air, moisture, and potential contaminants can increase the risk of spoilage. For this reason, many ketchup bottles include a recommendation to refrigerate after opening."

So in short:

Fridge recommended for quality, but if you're using it regularly, it'll probably be fine out of the fridge.