The Signature Growth conjecture for positive links

For a link $L$, let $n'(L)$ denote its number of components and let $\chi(L)$ be its Euler characteristic. Define

$$\chi'(L):=n'(L)-\chi(L).$$

Let $\sg(L)$ denote the signature of $L$. Signature Growth conjecture.

$$\liminf_{n\to\infty}\,\min\{\,\sg(L)\,:\,L\text{ is a positive link},\ \chi'(L)=n\,\}=\infty.$$

Equivalently, for every fixed signature value, only finitely many values of $\chi'(L)$ should occur among positive links. The conjecture formalizes the expectation that the signature of positive links grows as their negated Euler characteristic increases; the paper describes it as suggestive but difficult to approach, with supporting evidence from average-signature results.