Cuntz's positivity conjecture for exterior-power fusion algebras

Let $d\geq 1$, let $1\leq n\leq d$, and let $I_{n,d}$ be the set of strictly increasing $n$-tuples $(i_1,\ldots,i_n)$ with $0\leq i_1<\cdots<i_n<d$. Set $\xi=\exp(i\pi/d)$, $\zeta=\xi^2$, and let $S=(\zeta^{ij}/\sqrt{d})_{0\leq i,j<d}$. For $p\in\mathbb Z$ and $a,b,c\in I_{n,d}$, define

$${}_pN_{a,b}^c=\sum_{k\in I_{n,d}}\frac{(\bigwedge^nS)_{a,k}(\bigwedge^nS)_{b,k}\overline{(\bigwedge^nS)_{c,k}}}{(\bigwedge^nS)_{i^{(p)},k}}.$$

These are integers, and hence structure constants of a $\mathbb Z$-algebra. Cuntz's positivity conjecture. Suppose that $1<n<d$. If $n$ and $d$ are not both even, there exist signs $(\sigma_a)_{a\in I_{n,d}}\in\\{\pm1\\}^{I_{n,d}}$ such that ${}_pN_{a,b}^c\sigma_a\sigma_b\sigma_c\geq0$ for all $a,b,c\in I_{n,d}$. If both $n$ and $d$ are even, then for every such choice of signs there exist $a,b,c\in I_{n,d}$ for which ${}_pN_{a,b}^c\sigma_i\sigma_j\sigma_k<0$; nevertheless, the absolute values of the ${}_pN_{a,b}^c$ define an associative $\mathbb Z$-algebra. This conjecture concerns positivity, up to signs, of the fusion structure constants associated with the exterior power of the cyclic Fourier matrix; the integrality is stated earlier in the paper, while the positivity and associativity assertions remain conjectural here.