Bondy's conjecture on long circuits generating the cycle space

Let $d\in\mathbb{Z}$, let $X$ be a vertex-3-connected graph with $f_0(X)$ vertices, and let $\delta(X)$ be its minimum degree. Write $\mathrm{Z}_1(X;\mathbb{Z}/2)$ for the cycle space of $X$ over $\mathbb{Z}/2$.

Bondy's conjecture. If

$$f_0(X)\geq 2d\qquad\text{and}\qquad \delta(X)\geq d,$$

then the set of all circuits of length at least $2d-1$ is a $\mathbb{Z}/2$-generating system of $\mathrm{Z}_1(X;\mathbb{Z}/2)$.

The conjecture proposes that sufficiently long circuits generate the entire cycle space of a sufficiently dense, vertex-3-connected graph. The source notes that Locke proved it under the additional assumption that $X$ is non-Hamiltonian or $f_0(X)\geq 4d-5$; the general statement remains open.