That total includes supers, though. There are 4,051 pledged and 714 supers, totaling 4,765. You need 2,383 to win and 2,026 for a majority of pledged delegates. So, while it is extremely likely for one of two candidates to get a majority of pledged delegates—which should end the race right there—it's unlikely that someone wins outright at this point with a majority of total delegates. That would require a lead of more than 714 pledged delegates.
Thus, if you were willing to poison your reputation with the other side's supporters, you could attempt to swing up to 714 supers in order to clinch. I'm not sure if this would be a "contested" convention though. It would be more like Bernie wtf stop what even is this.