Constant-order structure conjecture for Reshetikhin–Turaev theory

Let $p$ be a fixed prime with $p\geq 3$. Let ${\cal V}^I_p$ denote the integral TQFT, and let $\dov {\cal V}^{FN}_p$ denote the corresponding Fibonacci-type TQFT; the tensor product is taken $\frac{p-3}{2}$ times. Constant-order structure conjecture. The irreducible components of ${\cal V}^I_p$ can be found as irreducible components of

$$\bigotimes^{\frac{p-3}{2}}\dov {\cal V}^{FN}_p.$$

This conjecture concerns a proposed polynomial dependence between the integral and Fibonacci-type Reshetikhin–Turaev theories, motivated by the corresponding relation between their asymptotic dimension growth rates. The authors note that it can be verified for the genus-one mapping class group, whose representation is ${\rm SL}(2,\mathbb{Z})$; its status in general is not specified here.