The complement-of-Fano blow-up conjecture for $J_4$ in codegree squared density

Let $G$ be the iterated blow-up of the complement of the Fano plane. It is $J_4$-free and has codegree squared density at least $0.28$. The complement-of-Fano blow-up conjecture for $J_4$. The graph $G$ is the extremal example in $\ell_2$-norm, namely,

$$\sigma(J_4)=0.28.$$

The same construction had previously been conjectured to be asymptotically sharp in $\ell_1$-norm. The supplied text gives no resolution for the $\ell_2$-norm assertion.