Denti,
There is no information about the FPV in the airspeed unreliable checklist, however there is a big note which could explain while it is not entirely wise to trust it completely.
Since the FPV takes static system inputs any malfunction that includes the static system may lead to wrong indications.
But the 737 FCTM goes to great length to say how groovy the FPV is with unreliable airspeed
