Non-categorifiability conjecture for iterated integral fusion-ring extensions

Let $\mathcal{R}=\mathcal{R}_0$ be an integral fusion ring, and let $(\mathcal{R}_n)$ be the sequence obtained by iterating the integral extension construction of Proposition, starting from $\mathcal{R}_0$. A non-categorifiability conjecture asserts that there exists an integer $n$ such that $\mathcal{R}_n$ does not admit a complex categorification. The claim predicts that iterated integral extensions eventually produce a fusion ring outside the class of complex categorifiable fusion rings, although the supplied text gives no evidence resolving this assertion.