The thief's premise of doing a jewel heist to promote his new book is flawed. They don't just forgive the crime of burglary if the criminal returns the item even if he claims it was a publicity stunt.
When Neill (Peter Haskell) is getting into his "burglar clothes" in the dressing room, he briefly steps in front of a three-panel mirror. There's a crew member reflected in the left-side panel of the mirror as he's entering the frame, and again as he leaves it.
The building where the robbery occurs is different from the video where the television show is taping. There is no way for the perpetrator to move from rooftop to rooftop in the time allotted for the crime.