As said in the comments, I believe there might be a correlation, but I'm skeptical as to what the correlation means.
One of the main pillars of productivity is doing the most important thing first. Productive people might prioritize work as the most difficult thing to do, so they head straight to work first thing in the morning. Others may have young children and care to them first thing in the morning, with their work as second priority.
You can say that a lot of the successful top management can get to work at 5 AM, but that's because they're not the ones who have to send their kids to school. Besides, why stay at home frying eggs when you have a gourmet chef in the office?
The unproductive person goes to work with the intention of filling his side of the contract, i.e. working 8 hours a day. He does not try to put in a good 8 hours. Instead, he tries to take as much of it for selfish personal use, but at least show that he's "doing his job". He may come to work early, if there is an incentive to do so or disincentive to doing otherwise, but given the flexibility, he would rather not go to work at all and try to shave off as much time as possible.
The most productive people will start by tackling the difficult tasks first. His work often aligns with his life goals. He gets fidgety if he stays at home and does nothing in the morning, so will try to show up for work as early as possible and leave when he's done his best.
Then you have the professionals whose job does not align with their life goals, but are doing it for the money/other rewards. They keep strictly to their contracts, perhaps giving a little more than asked. They're very productive too, but more out of discipline than motivation.