Lug nut question for 16” TRD Pro Beatylocks

Hey guys, I’m probably way overthinking this but I’d like to get it right the first time.. I know the thread size is correct. I am almost positive I need them, but I want to be sure.. My questions are whether or not I need the “small diameter” version of these nuts and locks, and what the length needs to be? The diameter for the “small diameter” is .80” (which I roughly measured on my lug nuts and seems about right). The length is 1.36”.
Once again, I’m likely way overthinking this. As always, thanks for taking the time to read and give your opinion! Below are pictures of my wheels, the lug nuts, and the wheel locks I’m looking at.

 

If these are the same wheels, it says 12mm conical style
Under parts required it gives -
PTR27-35090

https://trdtoyotaparts.elmoretoyota.com/oem-parts/toyota-trd-pro-16-in-off-road-beadlock-style-wheel-ptr1835090


https://trdtoyotaparts.elmoretoyota.com/oem-parts/toyota-wheel-locks-trd-27-pc-set-ptr2735090

 

This right here!

 

Yes, the part number and links you listed is what I have now. My question is whether the Gorilla lug nuts I am looking at are long enough/narrow enough to work. I haven’t been able to find measurements for the oem lug nuts I have now.
The gorilla lug nuts and wheel locks I listed are also conical.

 

Maybe this will help.
The box in the picture shows some dimensions.
https://www.tacomaworld.com/threads/found-cheaper-solution-for-trd-beadlock-lug-nuts.415963/

 

Man, this was extremely helpful. Thanks for the link to this thread. I wouldn’t think length would be the issue after reading some of the thread. It seems the problem is some aftermarket lug nuts/lug nut spline tools being too large in diameter and rubbing on the edges of the openings when loosening/tightening.