Existence conjecture for non-trivial $F$-irregular graphs

Let $F$ be a connected graph of order at least $3$. An $F$-irregular graph is a graph in which no two distinct vertices have isomorphic $F$-neighbourhoods.

Existence conjecture. For every connected graph $F$ of order $3$ or more, there exists a non-trivial $F$-irregular graph.

The conjecture concerns the existence of irregular graphs relative to an arbitrary connected graph $F$. It is known for stars and complete graphs of order at least $3$, but the general case is not resolved in the supplied text.