The Keevash–Mubayi extremal uniqueness conjecture for $K_t^-$

For $t\geq 2$, define degenerate $3$-graphs $H_t$ by

$$H_2=([2], \{111,222,112,221\}),$$

and

$$H_3=([3], \{111,222,333,112,223,331\}).$$

For $t\geq4$, obtain $H_t$ from $H_{t-2}$ by adding vertices $t-1,t$ and edges $(t-1)(t-1)(t-1),ttt,(t-1)(t-1)t,(t-1)tt$. Let $G_t(n)$ be the complement of a balanced blow-up of $H_{t-1}$ on $n$ vertices. Write $K_t^-$ for the complete $3$-graph on $t$ vertices with one edge removed, and let $\operatorname{ex}(n,K_t)$ and $\operatorname{ex}_{K_t^-}(n,K_t)$ denote the corresponding extremal quantities.

Keevash–Mubayi conjecture. $G_t(n)$ is the unique, up to isomorphism, $3$-graph with $\operatorname{ex}(n,K_t)$ edges and $\operatorname{ex}_{K_t^-}(n,K_t)$ induced copies of $K_t^-$. The construction is known to be $K_t$-free, but the asserted simultaneous extremality and uniqueness are not established in the source.