Not yet but now that there's so much overlap between the two, it's possible one will try to undercut the other. When you're producing tens of million of devices, a small pricing advantage of one over the other can move the needle. The biggest strength of HDMI now is going to be how many other existing AV devices there are that only support HDMI.
Sure there’s overlap, and DisplayPort could mostly do what HDMI could do for the past 10 years (apart from minor differences where one standard eg supported 4K a but earlier).
Nevertheless HDMI stayed the main interface in the TV/Entertainment world. So I doubt this will change anytime soon
Screens will probably get new ports, but I seriously doubt HDMI ports on new screens will be removed before the VGA ports are. New TV's still have SCART connectors.