The eventual monotonicity conjecture for minimal subword entropy

For $k \geq 2$, let $\min S_{\mathrm{sw}}^{(k)}(n)$ denote the minimal subword entropy among words of length $n$. Eventual monotonicity conjecture. There is a value $N$ such that the function

$$\frac{\min S_{\mathrm{sw}}^{(k)}(n)}{n}$$

is increasing for $n \geq N$.

The function is known experimentally not to be increasing for all $n$, so the conjecture asserts only eventual monotonicity. Its general status is open.