The door does not lock so the cycle never begins, the timer does not count down. The door latches shut but pressing the start button does not start the cycle which should then lock the door.

This is most likely the problem and correct solution. I was unable to remove the handle to remove the stem and replace the washer. I took the entire faucet to the hardware store and they were also stumped. I ended up replacing the entire faucet.

I have submitted a ticket with them but have not heard back.

@WarLoki is there a more appropriate StackExchange to ask this question?