Answer:
I believe this that the chef does not begin to believe his own lie, he instead constantly denies it. ”Let me go home with you . I'll be a father to your son, and ill warm your bed, and I'll keep you safe. “This demonstrates that he never came to a conclusion to believe the lie.
although, on the other hand, it also states this: towards the end of the story "The briefcase" the chef's begins to believe in his own lie as he starts referring to the time he spent as a chef as "in another life time". When the professor's widow wants to get him arrested for the identity theft of her husband he says “The woman does not see how she is murdering her husband, right here in the post office lobby.” This shows that the desperation to save his own life has forced him to adopt extreme measures. He can go to any length to prove that he is the real professor, even killing his own identity.
i hope this helps. Let me know.