I can't tell if that's a cut or a brush burn. Either way i don't think hair is ever going to grow there again. there might not be a visible scar on the skin when it heals, but i doubt your gonna see any hair grow back.
Paliminos are light horses, and if they get a bad scrape the hair often grows back in dark (if it grows in at all). A friend of mine has a halter show stallion and he ripped himself open on a fence. She used cold bacon grease on the scab twice a day, the hair grew back quickly and was not darkened. It doesn't hurt to try... of course, there is the difficulty in getting the dog not to lick it incessently...
Past that, I don't know anything else to try that hasn't already been suggested.