Existence and non-existence conjecture for ${\cal A}\cT_{n,\omega_U}$

Let ${\cal A}\cT_{n,\omega_U}$ denote the fusion category associated with the construction considered in the paper, indexed by an integer $n$ and a parameter $\omega_U$. The preceding results show non-existence for $4\leq n\leq 10$ and force $\omega_U=1$ whenever the category exists in the cases studied. Existence and non-existence conjecture. The technique used for the preceding uniqueness and non-existence theorems should show that ${\cal A}\cT_{n,\omega_U}$ exists only if $\omega_U=1$ for all $1\leq n<\infty$, and that ${\cal A}\cT_{n,\omega_U}$ does not exist for all $4\leq n<\infty$. This would extend the computed results to all values of $n$, ruling out the construction from $n=4$ onward and leaving only the parameter value $\omega_U=1$ as a possibility.