I am not worried about Chambers playing slow in June. Even with typical weather, the sandy base and fescue means you can get really firm and fast. Sure, the greens won't be 13 or 14, but it was never going to be that speed (or should be) anyway.
Even during US Am, after the stroke play, they dumped many inches of water to soften up the course for the match play because it was too firm. It just won't be as brown in June, but it will be plenty fast.
The only thing I worry about is the density of fescue. During US AM, they rolled and rolled those greens so the putts rolled true even though there was as much sand as grass, but I am not sure that will fly in June with more moisture. Unless they can greatly increase the grass density on greens, there will be much complaining from the pros at US Open.