Yeah, that is an outright lie.
The only exception being for those who start T rather late in life, where their larynx has already ossified or is in the process of becoming bone. In that instance the cartilage, not being cartilage anymore, won't grow and the voice wouldn't be able to change all that much as a result (the vocal chords will still stretch but would be trapped in a too-small larynx structure). In that instance, and only that instance, would that claim be mostly true. Otherwise, if it is still cartilage and not bone, it'll grow and it'll change a whole lot.
I know mine has gotten about three times its size already. I actually felt structures being pushed aside to accommodate it (especially in the first three months). If I moved my head and jaw a certain way they'd lock up and stress the ligaments in there, feeling too short and pulling. Still happening too. And it feels a bit uncomfortable squarely in that area in the second week after my dose (where I notice my voice consistently changing now, like clockwork).
Actually, NONE of my Oxford's fit me in my neck anymore because of this alone. They ALL choke me because they push back on it too much. They were all between 1/2-1" between my neck and collar before and now there's no room. I was a 17" neck before too. Its going to be really hard to find a 20" neck but in a medium or large trunk... seriously. Certainly not going to be a department-store find.
I still can't use my voice without it breaking. I mean like, it'll just stop making noises altogether at even medium ranges so it's still growing and changing and probably won't stop for a few years yet.
Anyway, bonafide BS from the NHS (so long as you're not or aren't totally ossified in your larynx cartilage).