Looking at the FCTM, following the QRH and then doing nothing may put you deep in the do-do.
"Apply the initial actions of the UNRELIABLE SPEED INDICATION QRH procedure from memory as they quickly provide a safe flight condition in all phases of flight and aircraft configuration."
"After applying the QRH procedure, and when the aircraft flight path is stabilised, attempt to identify the faulty ADR(s). Once the faulty ADR(s) has/have been positively identified, it/they should be switched OFF. This will trigger the corresponding ECAM procedure, which should be applied."
CL thrust and 5 degrees at say 2000ft below Max Alt in convective WX, that does not leave you long to troubleshoot ! :o
This may be one of those situations where blindly following the QRH may have caused more trouble than good.