My 2007 Explorer Sport Trac has this exact same part and problem, possibly exacerbated by a leaky third brakelight that previously caused rainwater to run into the subwoofer area. The brake light seal was easily fixed with some silicone, but I've been bothered by the subwoofer regularly cutting out for about a year.
I was getting ready to order a replacement subwoofer amp when I came across this post. Touching up the solder on those caps seems to have done the trick to fix this annoying problem. Thanks!!!