The continuous-limit correspondence between differential and q-difference spectral types

Consider an $m \times m$ Fuchsian differential system

$$\frac{dY}{dx}=\sum_{j=0}^n\frac{\mathcal{A}_j}{x-u_j}Y,\qquad u_0:=0.$$

Let $\mathcal{P}_j=m^j_1,\ldots,m^j_{\ell_j}$ for $j=0,\ldots,n,\infty$ be the partition of $m$ corresponding to $\mathcal{A}_j$, so its spectral type is $\mathcal{P}_0;\ldots;\mathcal{P}_n;\mathcal{P}_\infty$. Consider an $m\times m$ linear $q$-difference system

$$Y(qx)=A(x)Y(x),\qquad A(x)=A_nx^n+\cdots+A_0,$$

with spectral type $S_0=\mathcal{P}_0$, $S_\infty=\mathcal{P}_\infty$, and

$$S_{\mathrm{div}}=m^1_1,\ldots,m^1_{\ell_1},\ldots,m^n_1,\ldots,m^n_{\ell_n}.$$

After changing the dependent variable by $Z(x)=f(x)Y(x)$ for a suitable scalar function $f(x)$, the system becomes

$$\frac{Z(x)-Z(qx)}{(1-q)x}=\frac{1}{(1-q)x}\left\{I_m-\frac{f(qx)}{f(x)}A(x)\right\}Z(x).$$

Continuous-limit correspondence. If, as $q\to1$, the zeros $\alpha^k_j$ of $\det A(x)$ corresponding to $m^k_j$ satisfy

$$\alpha^k_j\to u_k\qquad (j=1,\ldots,\ell_k),$$

then the linear $q$-difference equation can tend in the continuous limit $q\to1$ to the given Fuchsian differential equation. This proposes a correspondence between the spectral types of the two systems under the continuous limit. The claim concerns the proposed degeneration from the specified $q$-difference system to the Fuchsian differential system; the source provides no resolution or further evidence of its status.