When posing a "do I/don't I refrigerate after opening" question, it might be helpful to tell where you live as an additional bit of information.
What you can successfully not refrigerate in Wisconsin just might be a big problem in St. Petersburg, FL, New Orleans, Hot-Lanta (you get the picture).
This country is BIG. That factor alone will cause "your mileage to vary" with regard to many foodstuffs.
WAAAAY down hee-yah in USDA zone 10, if seems like once I walk to the pantry to get the peanut butter, by the time I get back to the plate, the bread is starting to mold.
And I keep my bread in the freezer.