They absolutely can be sharpened, the "cannot" line is wrong, and the reason it persists is that the method is unlike anything you do on a stone.
Start by understanding the geometry, because it makes everything else obvious. A serrated edge is bevelled on one side only. The other side is flat. Each scallop is a tiny curved edge, and the points between them are what actually start the cut.
So the work is:
One scallop at a time, on the bevelled side only. You need something round that fits inside the scallop — a tapered sharpening rod is the purpose-made tool, but a round or half-round needle file, a chopstick wrapped in wet-and-dry paper, or a ceramic rod all work. The taper matters because your scallops are not all the same size; you slide the rod in until it seats and use that spot.
Match the existing angle, do not invent one. The bevel is already ground and you are refreshing it, not reshaping.
A few strokes each, all in one direction, working along the blade. You are raising a small burr on the flat side.
Then lay the flat side dead flat on a fine stone and take the burr off with two or three very light passes. Flat means flat — no angle at all. This is the step people miss, and skipping it is why a home-sharpened serrated knife often cuts worse afterwards.
Twenty minutes for a bread knife, and it will be a different knife.