So I spent some time on this issue and found that adding below code in your functions.php in your theme folder fixes custom date issue. Now it actually updates published date on the post. http://www.imageno.com/ytc6wy3yfrjepic.html
However, update is still not working. any suggestions??
We have added a new feature for this date field. If a user select a date from the custom field then this date will be the published date of that post. We have added this option too. So, you have to wait till our next release.