AndroidCommunity asked LG if they plan on introducing an LG G2 GPE anytime soon, and it seems that right now they have no plans to do that:
As of right now, LG doesn't have any plans for a Google Play Edition. The focus for the G2 is penetration through carriers. That however could change down the road." -Ken Hong, LG PR
It's disappointing to see that LG doesn't seem committed to the Google Play Edition program, even though it's the Nexus 4 and the collaboration with Google that even led to them being worth of attention in the media today, because before that people almost forgot LG existed, especially after the LG Optimus 2X blunder.
LG must be very happy with themselves that they got the support of the 4 major carriers in US, even though that comes with some major limits, and they are probably thinking that they don't really need the "core" Android users to get excited about their phones anymore and create buzz for them, because the carriers have their backs now, and will promote the phones for them.
In LG's place, I'd be very wary about trusting the carriers. The carriers will dump their phone as soon as there is something better around the corner, and especially if the people don't seem very excited about the phone, even after all the marketing they will be doing for it.
LG does say that things could change down the road, but that is code word for saying they aren't even thinking about it right now, but in case the public is very outraged about it, then maybe they will. That's why I think it's important to demand from all of these manufacturers to create Google Play Edition phones, and then vote with your wallet and buy those phones.
The Android community has one chance, and one chance alone to make this program work. If even one generation of these phones doesn't turn out very well in terms of sales, or if people don't seem to want them, then we'll probably lose the program forever, and the OEM's will get back to making their phones as customized as possible.
If you think stock Android phones are important, then go to LG and tell them to support the Google Play Edition program with as many phones as they can. The manufacturers may give Google a shot with this program, if they're asking them, but they're not going to continue it if the consumers stop demanding them and buying them, and some OEM's like LG won't even bother supporting it.