Yes they most certainly will. The rule for setting the light is anytime the system goes 1.5 times the FTP (federal test procedure) limits. For many issues the light won't come on till the second time a failure occurs. So in that case all single failures set what some manufacturers call pending codes, still readable.

Some fault codes are for things that don't effect emissions and never set a light.

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Steve Brotherton
Continental Imports
Gainesville FL
Bosch Master, ASE Master, L1
33 years MB technician