Finiteness conjecture for quasi-alternating links of fixed Jones-polynomial breadth

Let $b$ be an integer, and consider quasi-alternating links whose Jones polynomial has breadth $b$, where the breadth is the difference between the highest and lowest exponents occurring in the Jones polynomial. The finiteness conjecture. There are only finitely many quasi-alternating links with breadth $b$.

This is motivated by the analogous finiteness result for alternating links, for which the breadth of the Jones polynomial equals the crossing number. The conjecture remains open, although it has consequences for obstructions to a link being quasi-alternating.