Step 4: Confirmation of diagnosis
Assuming you have found a faulty core, get yourself a new one:
Volvo 9144221 Heater Core/matrix/radiator (an OEM part costs circa US$150 or £100). Order new o-rings while you are at it.
Ft the new core into the white plastic using those four torx screws, and the o-rings on the pipe stubs.
I didn't check my o-rings and all was OK, but if for some reason they went missing or were damaged you would find that your system could leak at this joint. Don't take this risk.