Existence conjecture for link-irregular tournaments

Let a link-irregular tournament be a tournament in which the directed links of every pair of distinct vertices are non-isomorphic. Link-irregular tournament conjecture. A link-irregular tournament exists on $n$ vertices if and only if

$$n\geq 6.$$

The paper reports explicit constructions for orders up to $8$ and computational verification up to order $100$, but the general existence claim remains open.