The iterated-blow-up conjecture for $\{K_4^{3-},C_5\}$

Let $G$ be the iterated blow-up of an edge, obtained from a complete balanced $3$-partite $3$-graph by inserting a complete balanced $3$-partite $3$-graph recursively in each part. This $3$-graph is $\{K_4^{3-},C_5\}$-free and gives the lower bound $\sigma(\{K_4^{3-},C_5\})\geq 1/13$. The iterated-blow-up conjecture for $\{K_4^{3-},C_5\}$. The graph $G$ is the extremal example in $\ell_2$-norm; in particular,

$$\sigma(\{K_4^{3-},C_5\})=\frac{1}{13}.$$

The construction is described as the current best lower-bound example, and no matching upper bound or resolution is supplied.