The uniform partially-intersecting partition-system conjecture

Let $\mathcal{U}^n_k$ be the family of uniform $k$-partitions of an $n$-set. Two partitions are partially $t$-intersecting if some class from each has intersection of size at least $t$.

Uniform partial-intersection conjecture. Let $k,c,t$ be positive integers with $t\leq c$ and $n=ck$. If $\mathcal{P}\subseteq\mathcal{U}^n_k$ is partially $t$-intersecting, then

$$|\mathcal{P}|\leq\binom{n-t}{c-t}U(n-c,k-1).$$

Moreover, equality holds if and only if, up to a permutation of $[1,n]$,

$$\mathcal{P}=\{P\in\mathcal{U}^n_k:[1,t]\subseteq P_i\text{ for some }P_i\in P\}.$$

The conjecture generalizes the preceding partially 2-intersecting problem from arbitrary partition systems to uniform partition systems.