Turán's conjecture for the Turán density of $K_4^-$

Let $K_4^-$ be the $3$-graph on four vertices with three edges, and let $\pi(K_4^-)$ denote its Turán density.

Frankl–Füredi–Razborov conjecture.

$$\pi(K_4^-)=2/7.$$

The conjectured value comes from the balanced iterated blow-up of $H_6$. The source gives an upper bound close to $2/7$, but not equality.