Bobkov–Houdré–Tetali's optimal functions conjecture for odd cycles

Let $C_n$ be a cycle, let $X$ be an optimal function on $C_n$, and let $d$ denote graph distance. Then $X$ is an optimal function in the subgaussian sense described in the source.

Bobkov–Houdré–Tetali conjecture. There exists a vertex $x_0\in V(C_n)$ such that, for every $v\in V(C_n)$,

$$|X(v)-X(v_0)|=d(x_0,v).$$

Bobkov, Houdré, and Tetali proposed this characterization for optimal functions on odd cycles, and Sammer and Tetali repeated it. The supplied text does not state whether the conjecture has been resolved.