We’re not both right, I am 
I’m being cheeky and mean that as tongue-in-cheek as is possible.
18yrs flying commercially & currently fly the 787. I promise you, pushing the nose down after main gear contact would almost certainly lead to the nose gear being badly damaged with a fair chance of it collapsing entirely. You’d be surprised how flimsy the nose gear is (relatively speaking) in comparison to the the mains.
The PMDG 737 behaves pretty much exactly as per the real aircraft.
Hope that clears it up.
Just to note, having read the post above this, that once the nose gear has been flown on to the runway nose down elevator is often applied to ensure good directional control, something that is vitally important in a strong crosswind or on a contaminated runway.
To be clear, in replying above I am talking about not applying nose down elevator to lower the nose wheel on to the runway immediately after the main gear has made contact. With the spoilers deploying and the autobrake / reversers kicking in all the momentum is bringing the nose down pretty rapidly. The spoilers are phenomenally powerful and dump lift almost instantaneously, so adding to that with forward elevator is an absolute no-no. On the contrary, you often almost need a second flare to actually fly the nose on to the runway. The golden rule is that you are still flying the aircraft until you’re at a safe taxi speed.