Scheduled posts and timestamps

I have a question regarding scheduled posts and timestamps. I maintain a WP blog for a company where, ideally, only some people are supposed to actually publish posts. Others can prepare the post, but supposedly only the limited set of users can publish the posts.

However, we sometimes run into issues with scheduled posts: sometimes, the people who are supposed to publish posts do so too late. The problem is basically this:

a) Person writes a post, sets the timestamp when he’d like it to appear. (Let’s say 3PM).
b) Publishers log into the blog admin panel late. Let’s say 4PM. They hit the publish button.
c) The post appeared on the blog at 4PM, but has a timestamp of 3PM.

Right now we’re fixing this problem manually, but is there a way to fix this so it’s no longer a problem?