The q-index characteristic-polynomial conjecture for braided orbits

Let $mu_i$, $1\le i\le p$, be roots of the polynomial ${\cal CH^{\chi}}_{(1)}(t)$. For integers $k_1,\dots,k_p\ge 0$ with $k_1+\dots+k_p=m$, define

$$\xi_p(k_1,\dots,k_p)=\sum_{s=2}^{p}q^{k_1+k_2+\dots+k_s-m}[k_s]_q[k_1+k_2+\dots+k_{s-1}]_q.$$

The q-index characteristic-polynomial conjecture. For every $m\ge 2$, the polynomial ${\cal CH^{\chi}}_{(m)}(t)$ has degree

$$\operatorname{deg}({\cal CH^{\chi}}_{(m)}(t))={m+p-1\choose m},$$

and its roots are given by

$$q^{m-1}\mu_{k_1\dots k_p}(m)=\sum_{i=1}^p\frac{[k_i]_q}{q^{m-k_i}}\mu_i+\xi_p(k_1,\dots,k_p)\hbar,$$

where $k_i\ge 0$ and $k_1+\dots+k_p=m$.

This conjecture proposes an explicit description of the characteristic-polynomial roots governing the q-index for higher-rank braided orbits. The paper presents low-dimensional computations supporting it, while the proof of the corresponding general formula remains open.