Simplicity conjecture for the zeros of the adjacency function

For every $\omega>0$ and every integer $l\geq0$, let

$$\zeta_l(\omega,\mu)=\xi_l\left(\frac1{4\omega^2}-\mu^2,\mu\right),$$

where $\xi_l$ is the functional equation associated with entire Heun solutions and adjacencies. Simplicity conjecture. The zeros of $\zeta_l$ are simple; equivalently,

$$\frac{\partial\zeta_l}{\partial\mu}\neq0$$

at every zero of $\zeta_l$. This conjecture concerns the local regularity of the adjacency loci; the source gives no resolution evidence.