Balogh–Clemen–Lidický's exact $cell_2$-norm Turán conjecture for $K_5^3$

Let $\mathcal{H}$ be an $r$-graph, and let $d_{\mathcal{H}}(e)$ denote the codegree of an $(r-1)$-set $e$. Define

$$\lVert\mathcal{H}\rVert_{2}=\sum_{e\in \binom{V(\mathcal{H})}{r-1}}d_{\mathcal{H}}(e)^2.$$

For an $r$-graph $\mathcal{H}$, let

$$\mathrm{ex}_{\ell_2}(n,\mathcal F)=\max\{\lVert\mathcal H\rVert_2:|V(\mathcal H)|=n\text{ and }\mathcal H\text{ is }\mathcal F\text{-free}\}.$$

Given a balanced partition $[n]=V_1\cup V_2$, let $\mathbb{B}_n$ be the complete bipartite $3$-graph consisting of all triples intersecting both parts. Balogh–Clemen–Lidický's conjecture. There exists $n_0$ such that for every $n\ge n_0$,

$$\mathrm{ex}_{\ell_2}(n,K_5^3)=\lVert\mathbb{B}_n\rVert_2,$$

and $\mathbb{B}_n$ is the unique $K_5^3$-free $3$-graph on $n$ vertices attaining this value. The $ell_2$-norm Turán density is already known to equal $5/8$; the conjecture asks for the exact extremal value and uniqueness for all sufficiently large $n$.