I haven't heard of any other tests with that beastly Sony computer CRT, so I'm not sure how much luck you'll have getting it to work...
It should work fine with the 37w3 via VGA, when I had mine it had no problem accepting 1080p via VGA from the PC, and it's bigger brother (the 42w2) doesn't have any problems with the 360 either.
I'm actually reading reports that some people are getting the 42w2 to accept 1080p through component (even though specs contradict this), but I don't think 37w3 owners have had as much luck with it.