Large Inducibility Conjecture for nontrivial graphs

For a finite graph $H$, let $\operatorname{ind}(H)$ denote its inducibility, and let $K_{|H|}$ and $\overline{K}_{|H|}$ be respectively the complete and edgeless graphs on $|H|$ vertices. The Large Inducibility Conjecture.

$$\limsup \left\{\operatorname{ind}(H): H \notin \{K_{|H|}, \overline{K}_{|H|}\} \right\} = 1/e.$$

The conjecture is presented as an analogue for graph inducibilities of the Edge-statistics Conjecture and would be implied by it. The known lower bounds include nontrivial graphs with inducibility approaching $1/e$, while the matching universal upper bound remains open.