Critical points of functions, $\mathfrak{sl}_2$ representations, and Fuchsian differential equations with only univalued solutions

Let a second order Fuchsian differential equation with only univalued solutions have finite singular points at $z_1,\dots, z_n$ with exponents $(\rho_{1,1},\rho_{2,1}),\dots,(\rho_{1,n}\rho_{2,n})$. Let the exponents at infinity be $(\rho_{1,\infty},\rho_{2,\infty})$. Then for fixed generic $z_1,\dots, z_n$, the number of such Fuchsian equations is equal to the multiplicity of the irreducible $\mathfrak{sl}_2$ representation of dimension $|\rho_{2,\infty}-\rho_{1,\infty}|$ in the tensor product of irreducible $\mathfrak{sl}_2$ representations of dimensions $|\rho_{2,1}-\rho_{1,1}|,\dots,|\rho_{2,n}-\rho_{1,n}|$. To show this we count the number of critical points of a suitable function which plays the crucial role in constructions of the hypergeometric solutions of the $\mathfrak{sl}_2$ KZ equation and of the Bethe vectors in the $\mathfrak{sl}_2$ Gaudin model. As a byproduct of this study we conclude that the set of Bethe vectors is a basis in the space of states for the $\mathfrak{sl}_2$ inhomogeneous Gaudin model.