Higher Cayley–Hamilton conjecture for 1-generic noncommutative orbits

Let ${\cal L}_{{\hbar},q}^{\chi}$ be a 1-generic quantum orbit defined by the stated relations, where $p$ is the symmetry rank of the Hecke $R$-matrix. Let $L_{(m)}$ be the matrix defined by the symmetrization construction, and let $(k_1,\dots,k_p)$ range over partitions of $m$ with $k_i\geq 0$ and $k_1+\dots+k_p=m$. Define

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

Higher Cayley–Hamilton conjecture. On ${\cal L}_{{\hbar},q}^{\chi}$, the matrix $L_{(m)}$ satisfies

$${\cal CH}_{(m)}^{\chi}(L_{(m)})=0,$$

where

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

and its roots $\mu_{\mathbf{k}}(m)$ satisfy

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

This conjecture proposes the higher Cayley–Hamilton identity and explicit spectral roots for quantum symmetric powers of a 1-generic noncommutative orbit. The source attributes it to earlier work, but the supplied text gives no resolution or further evidence of its status.