Bohman–Frieze–Mubayi–Pikhurko's codegree density conjecture for $F_{3,2}$

Let $F_{3,2}$ be the $3$-graph in the paper whose-free condition is equivalent to having independent joint neighbourhoods: for distinct vertices $x,y$, define

$$\Gamma(x,y)=\{z\in V(G):\{x,y,z\}\in E(G)\}.$$

A $3$-graph has independent neighbourhoods when every such joint neighbourhood is edge-free. Let $D_{A,B}$ be the one-way bipartite $3$-graph associated with a bipartition $V=A\sqcup B$, whose edges are all triples containing two vertices from $A$ and one from $B$. Let $T_{A,B,C}=D_{A,B}\cup D_{B,C}\cup D_{C,A}$ for a tripartition $V=A\sqcup B\sqcup C$.

Bohman–Frieze–Mubayi–Pikhurko's conjecture. The balanced construction $T_{A,B,C}$ is asymptotically optimal for the codegree problem for $F_{3,2}$; equivalently,

$$\gamma(F_{3,2})=\frac{1}{3}.$$

The construction has independent neighbourhoods and codegree approximately $n/3$ when the three parts are as equal as possible, giving the lower bound. The conjecture asserts that this lower bound is tight for the codegree density.