Conjectured interpolation formula for triangle-free subgraphs of random hypergraphs

For an integer $r\geq 4$ and a constant $x$ with $0\leq x\leq r$, let $G^r_{n,p}$ be the random $r$-graph on $n$ vertices in which each $r$-set is present independently with probability $p$, set $p=n^{-r+x}$, and define

$$f_r(x)=\lim_{n\to\infty}\log_n\mathbb{E}[\operatorname{ex}(G^r_{n,p},T^r)],$$

where $T^r$ is the loose triangle and $\operatorname{ex}(G^r_{n,p},T^r)$ is the maximum number of edges in a $T^r$-free subgraph of $G^r_{n,p}$.

Interpolation conjecture. For $\frac{3}{2}<x\leq 4$,

$$f_r(x)=\frac{3x+3}{5}.$$

The conjecture proposes that the upper bound obtained from the paper's container argument is the true value in the intermediate range. The theorem quoted immediately before it gives matching values for $0\leq x\leq\frac{3}{2}$ and $4<x\leq r$, but leaves this interval bounded rather than determined.