As to whether or not they will re-grow...maybe. I have seen barbels on cories regenerate, but I have also had cories where this did not happen. It depends upon the cause, and the extent of the damage. In the photo there seems to be a portion of the barbel left so it may re-grow if it was worn down. However, if this is bacterial in origin, it likely won't.
As for the cause...I wouldn't think from the photo the barbels were nipped off, and the fish in this tank are not too likely to be responsible, though the ram could but in my experience rams that get annoyed at cories--which is common when feeding since both feed from the substrate--tend more to just poke at them to drive them away from "their" food.
That leaves being worn down or bacterial, or both as bacterial can set in when the barbels are worn down by something. I would suggest you remove the gravel and get a soft sand. It is not just the possible roughness, it is also the grain size. If you have ever washed several buckets of gravel you will know how it can wear your skin a bit, even though it may be smooth. Cories continually poke their barbels into the substrate looking for food; this is their normal constant behaviour. Over time this can wear down barbels when the grains of gravel are as large as here. Another benefit of sand is that cories can take up a mouthful and sift out food, expelling the sand through the gill slits; this is not possible with gravel. So if you keep cories in this tank, a sand substrate is best; play sand is safe and inexpensive.
The other problem about gravel this large is that bits of food can get caught down in it, and decomposition is different than it is in sand or a finer gravel, and this can lead to bacterial issues in general, not just for the cories.
Ijn their habitats all species of cory live over substrates of either mud or sand, or a mix, with or without a layer of dead leaves. Replicating the natural habitat is always a good way to aquascape. We don't want mud, so sand that is soft can work very well. It is fun to watch cories literally upend themselves, as they dig down into the sand, and it comes spurting out their gills when they pull their heads back out.
Byron.
