The $4.2$-density conjecture in $K_4^-$-free $3$-graphs

Let $4.2$ be the $4$-vertex $3$-graph with exactly two edges, and let $\pi_{4.2}(K_4^-)$ denote its maximum asymptotic induced density in a $K_4^-$-free $3$-graph.

The $4.2$-density conjecture.

$$\pi_{4.2}(K_4^-)=24/43.$$

The lower bound is attained by a balanced iterated blow-up of $H_6$; the source's numerical upper bound remains slightly above $24/43$.