Nearly all $k$-SAT functions are unate

We prove that $1-o(1)$ fraction of all $k$-SAT functions on $n$ Boolean variables are unate (i.e., monotone after first negating some variables), for any fixed positive integer $k$ and as $n\to \mathrm{\infty }$. This resolves a conjecture by Bollobás, Brightwell, and Leader from 2003.