STS extremal construction conjecture for the $(2,p)$-norm Turán problem of $F_5$

Let $F_5$ be the 3-uniform hypergraph considered in the $(2,p)$-norm Turán problem, and let $\rho_{2,p}(F_5)$ denote its asymptotic extremal value. For a Steiner triple system (STS) $\mathcal{S}$, let $\lambda_{2,p}(\mathcal{S})$ denote its associated $(2,p)$-norm density. STS extremal construction conjecture. For every $p \in (0,1/2)$,

$$\pi_{2,p}(F_5)=\max\left\{2 \cdot \lambda_{2,p}(\mathcal{S}) \colon \text{$\mathcal{S}$ is an STS}\right\}.$$

The conjecture proposes that the asymptotic extremal construction for the $(2,p)$-norm Turán problem of $F_5$ is a blowup of some Steiner triple system. The paper establishes the corresponding value for $p$ around points in ${k^{-1}: k\in 6\mathbb{N}^{+}+{0,2}}$, while the assertion for every $p\in(0,1/2)$ remains open.