Existence of a small asymmetric fractalizer

A fractalizer is a graph whose balanced iterated blow-ups maximize the density of induced copies of the graph, and a graph is asymmetric if its automorphism group is trivial.

Asymmetric fractalizer conjecture. There exists an asymmetric fractalizer on at most $9$ vertices.

The paper motivates this conjecture by noting that graphs with twins are not fractalizers and that the smallest nontrivial asymmetric graphs have $7$ vertices. It is presented as a weaker conjecture about where a smallest nontrivial fractalizer may occur; no resolution is given here.