Saturation isn't the only problem. There's also too much contrast. As you can see below, when the shadow details are brought out, highlights pay the price. It's possible to maintain some highlight detail, but a super-high res raw file would be required as a working source document. From a photographer's perspective, one way to combat the problem is to take the same photograph using fill-flash. But even that is no easy route, as you have to balance exposure as well as colour temperature.