The quantum-dimension criterion for braiding universality of anyons

Let $a$ be an anyon, and let $\dim(a)$ denote its quantum dimension. An anyon is braiding universal when braiding sufficiently many copies of it gives a universal set of quantum gates.

Quantum-dimension criterion. The anyon $a$ is braiding universal if and only if $\dim(a)^2$ is not an integer.

This conjecture is motivated by the contrast between Fibonacci and Ising anyons: the former is universal, while the latter is not, although both are non-degenerate. For models associated with quantum groups, the braid-group image is known to be infinite if and only if $\dim(a)^2$ is not an integer, providing strong supporting evidence; the stated equivalence with universality remains unresolved here.