The asymptotic maximum-appearance conjecture for distinct-letter words in two-dimensional grids

Let $w'$ be a word of length $k\ge 2$ whose letters are all distinct, and let $f(w',n,2)$ denote the maximum number of copies of $w'$ appearing in an $n\times n$ grid. Distinct-letter word conjecture.

$$f(w',n,2)\sim \frac{3}{k-1}n^2.$$

The paper notes that the known bounds place $f(w',n,2)$ roughly between $\frac{3}{k-1}n^2$ and $\frac{4}{k-1}n^2$, and suggests that the lower bound is closer to the truth; the asserted asymptotic equivalence remains open.