Asymptotic prevalence of vanishing Grothendieck rings

Determine whether the limit inferior, as the bound on structure constants tends to infinity, of the proportion of vanishing Grothendieck rings among rank-r Grothendieck rings admitting pivotal categorifications over the complex numbers tends to 1 as the rank r grows, where the rings have structure constants bounded by that bound.

Background

For integers r,B1, the paper defines \mathcal{G}{r,B} as the set of based-ring isomorphism classes of rank-r Grothendieck rings admitting a pivotal categorification over \mathbb{C} and having all structure constants at most B. It defines V{r,B} as the subset of vanishing rings and considers the limit inferior of the ratio of the sizes of these sets as B tends to infinity.

The question asks whether vanishing becomes asymptotically ubiquitous among such categorifiable Grothendieck rings as the rank increases. The paper reports that vanishing is frequent in the currently known examples of ranks four and five, but it does not resolve this asymptotic question.