On free minimal constant speedups violating continuous orbit equivalence in $p$-adic $\mathbb{Z}^d$-odometers of adding type

Let $\mathbb{Z}_p$ be the ring of $p$-adic integers with respect to a prime $p$ and let $d$ be a positive integer. For each $\mathbf{z}=(z_1, z_2,..., z_d) \in \mathbb{Z}_p^d$, let $T_{\mathbf{z}}: \mathbb{Z}^d \times \mathbb{Z}_p \to \mathbb{Z}_p$ be an adding-type $\mathbb{Z}^d$-action on $\mathbb