The 4/4 conjecture for permutation rational functions

Let $\mathbb{F}_q$ be a finite field, and let $N_{v,u}(q)$ denote the number of permutation rational functions $V(x)/U(x)$ over $\mathbb{F}_q$ with numerator degree $v$ and denominator degree $u$. The 4/4 conjecture. For all prime powers $q$,

$$N_{4,4}(q)=\frac{(q+1)q^2(q-1)^3}{3}.$$

The formula has been verified computationally for all primes $p\leq 47$ and concerns permutation rational functions of equal numerator and denominator degree.