Date syrup and unripe grape juice: fridge, or pantry?

Just what the OP says - I just bought some date syrup and some unripe grape juice for recipes I've been wanting to try. Once they are opened, do they need to go in the fridge, or can I store them in the pantry? The labels provide no clues.

I'd put them in the fridge just to be safe. Date syrup may be so low in water content that it's shelf-stable, but sometimes even syrups can go off in the pantry (I read that sorghum syrup, for instance, can be refrigerated indefinitely but can develop unpleasant flavors if left on the shelf) Cook's Thesaurus lists verjus (unripe grape juice) here; and says open containers should go in the fridge.