Hi, and thanks for dropping by!
The software remains the same. The network lock/unlock has nothing to do with being able to root it. Give this post some time, for others to chime in, but I don't think you'll be in luck with rooting it, for now.
Source for below Info:
http://androiding.how/galaxy-note-5-root-status/
Verizon Note 5 Root: Not Available
It’d be hard to tell if Verizon Note 5 would ever get root access, let alone how and when. But that doesn’t mean you should lose hope. Things like Towelroot and Ping Pong Root often come and go, and while they stay, they set everything right.
AT&T Note 5 Root: Not available
Ditto Verizon !
The reason for this trouble that Verizon and AT&T Note 5 has with root access is the locked bootloader on these devices which prevents any modifications to the system partitions, hence no possibility of installing a custom kernel.