Uniqueness conjecture for maximum partially 2-intersecting uniform partition families

A $(k,\ell)$-partition is a set partition of $\{1,2,\dots,k\ell\}$ with exactly $\ell$ blocks, each of size $k$. Let $S_{i,j}$ denote the canonical partially 2-intersecting family defined earlier in the paper. Uniqueness conjecture. For $k\geq 3$ and $\ell$ sufficiently large, the only sets of partially 2-intersecting $(k,\ell)$-partitions with size

$$\binom{k\ell-2}{k-2}u_{k,\ell-1}$$

are the sets $S_{i,j}$. The preceding theorem establishes the corresponding maximum-size bound; this conjecture asserts uniqueness of the extremal families for sufficiently large $\ell$.