The non-elementary abelian group obstruction to 2-designs

Let $G=\mathbb{Z}_{p^{t_{1}}} \oplus \mathbb{Z}_{p^{t_{2}}} \oplus\cdots \oplus\mathbb{Z}_{p^{t_{m}}}$ be a non-elementary $p$-abelian group, with some $t_i>1$. For an integer $k$ and an element $x\in G$, let $(G,\mathcal{B}_k^x)$ denote the associated incidence structure.

Non-elementary group conjecture. For any $(k,x)$ with $1\le k\le n-1$ and $x\in G$, the incidence structure $(G,\mathcal{B}_k^x)$ is not a $2$-design.

Computational verification via MAGMA suggests that non-elementary abelian $p$-groups fail to produce $2$-designs for all nontrivial parameter pairs, in contrast with the elementary abelian case. The conjecture excludes the trivial case $k=|G|$ and remains open.