The conjecture on existence of F-irregular graphs

Let $F$ and $G$ be graphs. The $F$-degree of a vertex $v$ in $G$ is the number of subgraphs of $G$ isomorphic to $F$ that contain $v$. A graph $G$ is $F$-irregular if any two distinct vertices of $G$ have different $F$-degrees.

Existence conjecture. For every connected graph $F$ on three or more vertices, there exists an $F$-irregular graph.

The conjecture generalizes the known existence results for complete graphs and stars. It was later strengthened by the assertion that infinitely many such graphs exist for every connected graph $F$ of order at least $3$; the general conjecture is solved here only for the class of graphs $F$ of diameter $2$.