The complete-partite graph cycle-space conjecture for K^{s,s-1}

Let $\mathrm{K}^{\widehat{s},s-1}$ denote the graph described in the paper, and let $\mathrm{Z}_1(\mathrm{K}^{\widehat{s},s-1};\mathbb{F}_2)$ be its cycle space over $\mathbb{F}_2$. Let $\mathcal{H}(\mathrm{K}^{\widehat{s},s-1})$ denote the set of Hamilton circuits of this graph, and let $\langle\mathcal{H}(\mathrm{K}^{\widehat{s},s-1})\rangle_{\mathbb{F}_2}$ be their $\mathbb{F}_2$-linear span.

The complete-partite graph cycle-space conjecture. For every $s\geq 4$,

$$\mathrm{Z}_1(\mathrm{K}^{\widehat{s},s-1};\mathbb{F}_2) = \langle\mathcal{H}(\mathrm{K}^{\widehat{s},s-1})\rangle_{\mathbb{F}_2}.$$

The claim would provide an infinite family of graphs with a degree-$2$ vertex in which every cycle is a symmetric difference of Hamilton circuits, showing that the corresponding implication in the paper does not hold without additional hypotheses. The source presents this as seeming to hold based on examples and calculations, rather than as an established result.