Tanaka's topological complexity conjecture for Khalimsky circles

Let $\mathbb{S}^1_{2^k}$ be the Khalimsky circle with $2^k$ points, where $k\geq 1$, and let $\operatorname{tc}(X)$ denote the topological complexity of a space $X$. Tanaka's conjecture.

$$\operatorname{tc}(\mathbb{S}^1_{2^k})=2$$

for every $k\geq 1$. This predicts a uniform value for the topological complexity of these finite spaces, extending the known computations for small Khalimsky circles.