The characterization conjecture for Toeplitz speedups

Let $(X,T)$ be a Toeplitz flow with period structure $(p_k)$, and let $(X,S)$ be a bounded speedup of $(X,T)$ with orbit number $c$. For the Kakutani–Rokhlin partition $\mathcal P(k)$, let $\pi_i^{(k)}$ denote the cyclic orbit permutation associated with tower $i$, and let $o_i^k(m)$ denote the orbit height in tower $i$ at label $m$. Characterization conjecture for Toeplitz speedups. The system $(X,S)$ is a Toeplitz flow if and only if, for all sufficiently large $k$, every $\pi_i^{(k)}$ is the same cyclic permutation of $\{1,2,\ldots,c\}$ and

$$o_i^k(m)=o_j^k(m)$$

for every $m\in\{1,2,\ldots,c\}$ and all towers $i,j$. The paper proves the stated conditions are sufficient and conjectures their necessity, while the supplied text gives no resolution of the full equivalence.