Continuous orbit equivalence for bounded speedups of free $\mathbb{Z}^2$-odometers

Let $(X_{\mathfrak{G}},\sigma_{\mathfrak{G}})$ be a free $\mathbb{Z}^2$-odometer, and let $\mathbf{S}:\mathbb{Z}^2\curvearrowright X$ be a minimal speedup of $\sigma_{\mathfrak{G}}$ with bounded speedup cocycle $\mathbf{p}$. Two actions are continuously orbit equivalent when there is a homeomorphism between their phase spaces mapping orbits continuously to orbits in both directions. The continuous orbit-equivalence conjecture. The $\mathbb{Z}^2$-odometer $(X_{\mathfrak{G}},\mathbf{S})$ is continuously orbit equivalent to $(X_{\mathfrak{G}},\sigma_{\mathfrak{G}})$. This is motivated by the example preceding the conjecture, where continuously orbit-equivalent odometers are not isomorphic; the claim is stated as an open problem, at least in the $d=2$ case.