A quick test-- rub 2 of the pearls together gently. Do they glide smoothly on each other, or is there a gritty feeling? Smooth = fake, usually.
That bare spot on the end pearl does look like what a peeling imitation pearl looks like, but the photo is small. If the pearls feel gritty, post a close-up of the end pearl, and any other blemishes that stand out on the pearls.