I have a $k$-partite $k$-uniform hypergraph $H$ with $V(G) = V_1 \cup...\cup V_k$ (each $|V_i|=n$ for $i \in [k]$), such that the minimum vertex degree $\delta(H) \ge Cn^{k-1}$ for a constant $C$. I need to find a subhypergraph $H'$ with a linear partite minimum co-degree $\delta_{k-1}^*(H') \ge C'n$ for some constant $C'$.
The partite minimum co-degree of $H$ is defined as $$\delta_{k-1}^*(H) = \min_{\substack{S \in {V(H) \choose k-1} \\ |S \cap V_i|\le 1}} d(S).$$ In other words, if you fix any $k-1$ vertices, one in each set $V_i$ for $i \in [k]$, then there are at least $\delta_{k-1}^*(H)$ vertices in the remaining part that form an edge with $S$.
My ideas: Perhaps fixing $k-1$ vertices and deleting any possible candidate vertex in the other part which forms less than $C' n$ many edges with them might work. But I worry I might end up deleting everything. Or perhaps some kind of probabilistic construction might work as well, but I'm not very sure.