The rainbow Turán threshold conjecture for expanded cliques

Let ${\mathrm{ex}}^{\sum}_r(n,k,K_\ell^{(r)+})$ denote the rainbow hyper-Turán number for the $r$-uniform expansion $K_\ell^{(r)+}$, and let $t_r(n,\ell-1)$ be the number of edges in the complete balanced $(\ell-1)$-partite $r$-graph on $n$ vertices. Rainbow Turán threshold conjecture. For integers $\ell\ge r\ge3$, there is a constant $k_1=k_1(r,\ell)\ge \frac{\ell^2-1}{2}$ such that, for sufficiently large $n$, when $k<k_1$,

$${\mathrm{ex}}^{\sum}_r(n,k,K_\ell^{(r)+})= \min\left\{k,\left(\binom{\ell}{2}-1\right)\right\}\binom{n}{r},$$

and when $k\ge k_1$,

$${\mathrm{ex}}^{\sum}_r(n,k,K_\ell^{(r)+})=k\cdot t_r(n,\ell-1).$$

The conjecture is motivated by the two regimes known for the graph rainbow Turán number; the intermediate range is stated to be undetermined in the source.