Czabarka's conjecture for partially intersecting partition systems

Let $\mathcal{P}^n_k$ be the family of $k$-partitions of an $n$-set. Two partitions are partially $2$-intersecting if some class of one and some class of the other have intersection of size at least $2$; a system is partially $2$-intersecting if every pair is so related. Let $S(n-1,k)$ denote the corresponding extremal quantity used in the source.

Czabarka's conjecture. If $n\leq2k-1$ and $\mathcal{P}\subseteq\mathcal{P}^n_k$ is a partially $2$-intersecting partition system, then

$$|\mathcal{P}|\leq S(n-1,k).$$

The conjecture is attributed in the source to Czabarka and is cited to Erdős and Székely. The bound is attained by the system of partitions having a class containing a fixed pair.