Existence threshold conjecture for regular link-irregular graphs

Let $G$ be a graph on $n$ vertices. It is link-irregular if $G[N(u)]$ and $G[N(v)]$ are non-isomorphic for every pair of distinct vertices $u,v$, where $N(u)$ denotes the neighborhood of $u$ and $G[N(u)]$ the subgraph induced by that neighborhood. A graph is regular if all vertices have the same degree. Existence threshold conjecture. There exists a regular link-irregular graph on $n$ vertices if and only if $n\geq 12$. The paper establishes that no regular link-irregular graph exists for $n\leq 9$, extending earlier nonexistence results for regularities $0,1,2,3,4$. The conjectured threshold at $12$ remains open.