Non-invariance of circular turning-angle distributions under sub-sampling

Let $f_\circ(\theta;\lambda_1,\ldots,\lambda_n)$ be a circular distribution, let $f^{\{r\}}_\circ$ denote the turning-angle distribution obtained after sub-sampling every $r$ steps, and let $\Lambda$ be the step-length distribution. Non-invariance conjecture. There is no circular distribution $f_\circ(\theta;\lambda_1,\ldots,\lambda_n)$, other than the circular uniform distribution, such that

$$f^{\{r\}}_\circ=f_\circ(\theta;\hat{\lambda}_1,\ldots,\hat{\lambda}_n),$$

where $r\geq2$ is an integer, regardless of the step-length distribution $\Lambda$. This conjecture seeks a precise characterization of how sub-sampling changes turning-angle distributions in correlated random walks; the preceding results establish related effects for specific cases, while the general claim remains unresolved.