Airspeed calculation and indication depends on valid altitude calculation. Altitude calculation does not need airspeed, except for fine trimming. If icing the pitot tubes clogged them up, there is no logical reason to kill their companion static data outputs, as appears to have happened with this plane. It would seem to take a lot more icing to clog the static ports at the same time.
The Air Data boxes must have condemned the validity of their altitude outputs as well as airspeed, or the TCAS would not have flagged failed. TCAS does not have an airspeed input. Hence, if the "Pitot tubes are defective" is true, the Air Data boxes are also lacking necessary partitioning in their outputs.
By design, faulty airspeed computation/indication should not affect altitude computation/indication.
If clogged pitot were a not so rare phenomenon, altitude, attitude and ground speed would be used to simulate airspeed, using the last valid airspeed as baseline. But more than one pitot failing at a time is "too rare to consider."
GB