They check for OEM converter during smog check in California and Colorado. You can’t register your car without a smog check. They have no way of knowing the converter was stolen unless it is was engraved with the previous car’s VIN.
So yes, that’s not only what I’m suggesting, it’s what’s actually happening.
Most stolen cats are cut out with a sawzall.
I don’t care how good you think your welding skills are, reinstalling a cat that was removed with a sawzall will leave telltale welds.