The improved ball-number bound for links

Let $L$ be a link. Its ball number $\mathsf{ball}(L)$ is the minimum number of non-overlapping solid balls needed in a necklace representation of $L$, and its crossing number $\mathsf{cr}(L)$ is the minimum crossing number among diagrams of $L$.

Improved ball-number conjecture. For every link $L$,

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

Moreover, equality holds if $L$ is alternating.

The paper proves the weaker upper bound $\mathsf{ball}(L)\leq 5\mathsf{cr}(L)$ and presents an explicit construction of necklace representations. The proposed improvement to $4\mathsf{cr}(L)$, together with the alternating-link equality statement, is not resolved in the supplied text.