The complete-intersection conjecture for uniform partition systems

Let $\mathcal{U}^n_k$ be the family of uniform $k$-partitions of an $n$-set, and let a partition system be $t$-intersecting when every two partitions have at least $t$ common classes. For $0\leq i\leq\left\lfloor\frac{k-t}{2}\right\rfloor$, define

$$\mathcal{P}_i(n,k,t)=\left\{P\in\mathcal{U}^n_k:\left|P\cap\{[1,c],[c+1,2c],\ldots,[(t+2i-1)c+1,(t+2i)c]\}\right|\geq t+i\right\}.$$

Complete-intersection conjecture. For $n\geq k\geq t\geq1$, if $\mathcal{P}\subseteq\mathcal{U}^n_k$ is $t$-intersecting, then

$$|\mathcal{P}|\leq\max_{0\leq i\leq (k-t)/2}|\mathcal{P}_i(n,k,t)|.$$

The source presents this as the partition-system analogue of the Ahlswede–Khachatrian theorem. It also conjectures uniqueness up to permutations of the ground set; the fuller candidate statement is recorded here as the principal claim.