Strong conjecture on rooted F-irregular graphs

Let $F$ be a connected graph of order $|F|\geq 3$, and let $r$ be a root of $F$. A graph $G$ is $(F)_r$-irregular if the rooted copies of $F$ with root corresponding to each vertex of $G$ occur in pairwise distinct numbers. The graph $K_{1,n-1}$ is the star on $n$ vertices, and its center is its unique vertex of degree $n-1$.

Strong conjecture on rooted $F$-irregular graphs. For every connected graph $F$ of order $|F| \ge 3$ and each root $r$ of $F$, unless $F=K_{1,n-1}$ and $r$ is the center of $K_{1,n-1}$, there exist infinitely many $(F)_r$-irregular graphs.

The excluded star-center case is impossible: no nontrivial graph is $(K_{1,n-1})_r$-irregular there. The paper proves the corresponding assertion for paths except for the central root of $P_3$, and establishes simultaneous rooted and ordinary path irregularity in the stated families; the general conjecture remains open.