Cycle-space spanning conjecture for highly connected graphs

Let $G$ be an $n$-vertex graph, where $n$ is odd. Write $\kappa(G)$ for its vertex-connectivity, $\alpha(G)$ for its independence number, $\mathcal{C}(G)$ for its cycle space, and $\mathcal{C}_n(G)$ for the subspace spanned by the Hamilton cycles of $G$. Cycle-space spanning conjecture. There exists a constant $c$ such that, whenever

$$\kappa(G) \geq c\alpha(G),$$

we have

$$\mathcal{C}_n(G)=\mathcal{C}(G).$$

This conjecture removes the logarithmic connectivity condition required by the theorem proved in the source. It remains open, while the source establishes the conclusion under stronger conditions involving $\max\\{\alpha(G),\log n\\}$ or $\alpha(G)^2$.