The ball number bound for nontrivial nonsplittable links

Let $L$ be a link. Its ball number $\mathrm{ball}(L)$ is the minimum number of spheres needed to construct a necklace representation of $L$, and its crossing number $\mathrm{cr}(L)$ is the minimal number of crossings among all diagrams of $L$. A link is nontrivial if it is not the trivial link and nonsplittable if its components cannot be separated by a sphere.

Ball number conjecture. For any nontrivial and nonsplittable link $L$,

$$\mathrm{ball}(L)\leq 4\mathrm{cr}(L).$$

Moreover, equality holds if $L$ is alternating.

The proposed bound would improve the previously stated linear upper bound $\mathrm{ball}(L)\leq 5\mathrm{cr}(L)$. The source does not provide evidence that the bound or the equality statement has been proved or disproved.