Positivity and periodicity conjecture for the WZW fusion-ring elements $W^{(a)}_m$

Let $I$ be the node set, let $t_a$ and $h^{\vee}$ be the parameters associated with the Dynkin diagram, and let $W^{(a)}_m$ be the elements of the WZW fusion ring defined from the unrestricted $Q$-system. For $a\in I$, let $\tau_a\in O(\hat{\mathfrak{g}})$ and $\sigma_a=e^{-2\pi i n (\tau_a\hat{\omega}_0|\rho)}$ be as in the paper's tables. Positivity and periodicity conjecture. For every $a\in I$, the following properties hold: (i) $W^{(a)}_m$ is positive for $0\le m\le t_a k$; (ii) $W^{(a)}_{t_a k-m}=\tau_a(W^{(a)*}_m)$ for $0\le m\le t_a k$; (iii) $W^{(a)}_{t_a k}=V_{k(\tau_a\hat{\omega}_0)}$; (iv) $W^{(a)}_{t_a k+1}=W^{(a)}_{t_a k+2}=\cdots=W^{(a)}_{t_a(k+h^{\vee})-1}=0$; and (v) $W^{(a)}_{m+nt_a(k+h^{\vee})}=\sigma_a^n\tau_a^nW^{(a)}_m$ for $0\le m\le t_a(k+h^{\vee})-1$ and $n\in\mathbb{Z}_{\ge0}$. These assertions predict positivity, a duality relation, vanishing beyond the boundary, and twisted periodicity for the $Q$-system solution in the WZW fusion ring. Positivity is established in the paper for types $A$, $B$, $C$, and $D$ in the range $0\le m\le\lfloor t_a k/2\rfloor$; the full collection of properties is presented as the main conjecture and is not resolved in the supplied text.