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

Let $K_4^-$ be the complete $3$-graph on four vertices with one edge removed, let $C_5$ be the forbidden $3$-graph, and let $\pi(K_4^-,C_5)$ denote the corresponding Turán density.

The $K_4^-$-density conjecture.

$$\pi(K_4^-,C_5)=1/4.$$

The lower bound comes from an iterated balanced blow-up of a $3$-edge, while the source gives an upper bound below $0.251073$; the conjectured equality is unresolved.