The obstruction conjecture for Toeplitz speedups

Let $(X,T)$ be a Toeplitz flow with period structure $(p_k)$, and let $c$ be a positive integer. A minimal speedup with orbit number $c$ means a minimal bounded speedup $(X,S)$ of $(X,T)$ whose orbit number is $c$. Obstruction conjecture for Toeplitz speedups. If

$$\gcd\left(c,\frac{p_k}{p_{k-1}}\right)>1$$

for infinitely many $k$, then $(X,T)$ has no minimal speedup with orbit number $c$ that is a Toeplitz flow. In particular, if $\theta$ is a constant-length left proper substitution, then $(X_\theta,T)$ has no minimal speedup with orbit number $c=|\theta|$ that is also a Toeplitz flow. This is one of the paper's concluding conjectures, and the supplied text gives no resolution.