I'm by no means experienced but have done several Solent CTA transits and found them to be very helpful
I'd give them a call nice and early, they'll probably give you a ATSOCAS and tell you to remain outside while they coordinate your clearance. If you've still not recieved it as you approach the boundary then go underneath and climb once you've gotten clearance - if that's what you want.
I bet they'd rather you be communicating in some way (or at least squawking 0011 - Solent listening squawk
http://airspacesafety.com/media/474/...quawk_2012.pdf) than pottering on your merry way without talking to anyone.