robin,
I went to the Gangnam Service Center, talked someone from SK Telecom's Data Services dept, and Apple's support here in Korea. I still don't have LTE.
I don't have an ARC card (yet), so actually a friend is helping me out with this as well.
The Apple Support and the SKT data person were trying to be helpful. They actually wanted me to try some stuff, and asked me to follow up with them because they want to know the answer.
I told them I had a "friend" who was able to get LTE on SKT with the same Verizon iPhone. They asked me for their IMEI or other identifiers so they could look it up to see if it was set up differently from mine. I told them I didn't have that information, and that my "friend" was no longer using SKT and wasn't on LTE anymore.
The lady at the service center just ran through her regular diagnostics and then told me that there was nothing more she could do. It was the phone's fault, apple's fault, but she had done everything. I persisted and told her that my phone's LTE works fine. I've reset all network settings. It is either my phone's settings or SKT's setup of my phone is faulty. She eventually called the data services where a more helpful girl tried to help me out. We did the troubleshooting, but eventually she came to believe my situation. She's contacted Apple support a few times. Even Apple Korea's support has a case number for me, but they are telling me Verizon's phone can't do LTE on their network, etc. Which is all not true.
According to Apple Korea, my Verizon iPhone 5 is locked or there is some carrier restriction that Verizon needs to free up. I will call Verizon (tonight / tomorrow) to figure that out.
Only other solution may be to try doing a restore on my iPhone to return it factory settings and see if that helps it set the "profile" etc correctly.
I am doing the restore right now -- figure I could do this before calling, because from my understanding Verizon's iPhone is FULLY unlocked. However, I do know that it is unlocked for gsm / sim usage but there are definitely carrier differences (I can't access the APN settings on Verizon's iPhone).
I don't want to be the guy with the iPhone 5 on a 3G connection. I will keep you posted. Hopefully I find a resolution and this can help some other foreigners getting their LTE up and running on their iPhone 5.