The complete-uniform-hypergraph host conjecture

Let $r,t,n$ be positive integers, and let $K_n^{(r)}$ be the complete $r$-uniform hypergraph on $n$ vertices, with $K_t^{(r)}$ defined analogously. Write $_{K_t^{(r)}}(k)$ and $_{K_t^{(r)}}^*(k)$ for the simple-host and multigraph-host extremal functions.

Complete-host conjecture. If

$$k=(K_n^{(r)},K_t^{(r)})+1,$$

then

$$_{K_t^{(r)}}(k)=_{K_t^{(r)}}^*(k)={n\choose r},$$

and the unique extremal graph is $K_n^{(r)}$. This conjecture extends the exact complete-host result beyond the values of $k$ immediately above the complete-hypergraph Turán number; the statement does not specify a resolution.