The upper-bound conjecture for primes between $n$ and $kn$

Let $k$ be a positive integer, and let $n$ be a positive integer. The upper-bound conjecture for primes between $n$ and $kn$. The number of primes between $n$ and $kn$ is at most

$$\frac{kn}{9}+k^{2}.$$

The source presents this as an empirically motivated upper bound based on plotting and curve fitting; no proof or resolution is supplied.