The $4.2$-density conjecture in $(K_4^-,C_5)$-free $3$-graphs

Let $4.2$ be the $4$-vertex $3$-graph with exactly two edges. Forbidding both $K_4^-$ and $C_5$, write $\pi_{4.2}(K_4^-,C_5)$ for the maximum asymptotic induced density of $4.2$.

The $4.2$-density conjecture.

$$\pi_{4.2}(K_4^-,C_5)=6/13.$$

An iterated blow-up of a $3$-edge gives the lower bound $6/13$, and the source's upper bound is only slightly larger.