The codegree-squared extremal conjecture for the Fano plane

Let $\mathbb{F}$ be the Fano plane, and let $B_n$ be the complete balanced bipartite $3$-graph on $n$ vertices. For a $3$-graph $H$, write $\operatorname{co}_2(H)$ for its codegree squared sum and $\operatorname{exco}_2(n,\mathbb{F})$ for the maximum of $\operatorname{co}_2(H)$ over $\mathbb{F}$-free $3$-graphs $H$ on $n$ vertices. The codegree-squared extremal conjecture for the Fano plane. There exists $n_0$ such that for all $n\geq n_0$,

$$\operatorname{exco}_2(n,\mathbb{F})=\operatorname{co}_2(B_n).$$

Furthermore, $B_n$ is the unique $\mathbb{F}$-free $3$-graph $H$ on $n$ vertices satisfying $\operatorname{co}_2(H)=\operatorname{exco}_2(n,\mathbb{F})$. This is the analogue, for codegree squared density, of the solved extremal result for the Fano plane in the ordinary edge-count setting; the asserted exact extremal and uniqueness statement remains open in the supplied text.