Jailbreak/flash iphone for LTE?

OK so I got an iPhone 5S from my friend that used the phone on Verizon and I'm currently ON THE NETWORK with T-Mobile. THE APN SETTINGS ARE THE SAME. I can make calls and texts but I don't have the LTE I need. A friend suggested I go to T-Mobile and ask them to flash the phone. Can they do that? Apple also told me that the phone might need a Internet unlock. The phone is already unlocked but idek what an Internet unlock is. The phone was given to me restored but I have not restored it since i got it.

Can T-Mobile flash the phone to work on the LTE?

Does Verizon still need to unlock a feature on the phone to get the 4G LTE?

ALSO, If I jailbreak the phone, will that work?

Won't work. The LTE bands are different.

APN Settings of Verizon and T-Mobile are not same, set it right and restart your phone.

LTE!= GSM.

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