The higher-rank square-modulus conjecture for transversal difference numbers

Let $p$ be an odd prime and let $r\geq 1$. Set

$$G=(\mathbb Z/p^2\mathbb Z)^r,\qquad H=p(\mathbb Z/p^2\mathbb Z)^r.$$

For a transversal $T$ of $G/H$, write $D(T)=T-T$ and define

$$\delta(G,H)=\min_T |D(T)|.$$

Higher-rank square-modulus conjecture. For every odd prime $p$ and every $r\geq 1$,

$$\delta\bigl((\mathbb Z/p^2\mathbb Z)^r,p(\mathbb Z/p^2\mathbb Z)^r\bigr)=(2p-1)^r.$$

The case $r=2$ is the central odd square-plane problem, for which the paper proves only a weaker lower bound. The asserted formula is suggested by the coordinate box and remains open in higher rank.