Has he always done WP? I'm just wondering because some horses that have a background in Dressage, or a lot of years doing Dressage also do this and it is encouraged. Is he putting all of his weight on the bit causing a lot of pain in your back, or is he just constantly wanting to feel a tight rein? In dressage they want them to "seek the bit" or basically seek to always have rein contact. If this is the case, the horse will sadly always prefer this as this is where it's comfort zone is and where it gets it's confidence.
If it is really putting all of it's head weight on you, or you know for a fact it does not have Dressage training, then I agree with the exercises above :)
I am a bit mistyfied about this generalization. I have not had that to be my expereience; that horses trained in dressage will try to barg their heads down and lean on the rein way forward and down. I have seen dressage horse used to a very strong, constant contact, by not the kind of barging down the OP is discriginbg. That seems to me to be more like an old school hrose, or dude trail horse , who is immediately assuming someone is going to reef on his mouth.
you don't get a horse's head up by pulling it up. you get them to lift it themselves, by making them move forward , with a good surge. they will automatically lift their heads. you then give them a second's of not rein contact, then pick up your reins slowly and if they dive downward on the bit, goose them back up to move faster and they'll pick up their head. Again, don't snatch up the reins tight the instant they raise their heads, but take up your desired contact slowly and with gentleness. If they bring their head up as a result of you goosing them forward, and you immediately snatch the slack out of the rein, they will be punished for doing the right thing.
And, if you ease into contact and the hrose accepts the contact and flexes at the poll, then reward him with a bit of slack on the rein. you will be teaching him that head up, flexed is the place that is the best.