The h-vector form conjecture for odd cycle toric rings

Let $C_{2s+1}$ be the cycle graph on $2s+1$ vertices, and let $K[C_{2s+1}]$ denote its toric ring. Write its $h$-vector as a sequence with entries $h_1,h_2,\ldots,h_{s-1}$ appearing in the indicated positions. The h-vector form conjecture. The $h$-vector of $K[C_{2s+1}]$ is of the form

$$(1, h_1, h_2, h_3, \ldots, h_i, \ldots, h_{s-1}, h_{s-1} + (-1)^{s-1}, \ldots, h_i + (-1)^{i}, \ldots, h_3 - 1, h_2 + 1, h_1, 1).$$

This conjectural form describes the asymmetry of the $h$-vectors of toric rings of odd cycles. The preceding examples for $C_7$, $C_9$, and $C_{11}$ provide computational evidence, but the supplied text does not state whether the conjecture has been proved or disproved.