Algebraicity of the radial symmetry operators in the n-body problem

Let $\rho_{ij}=r_{ij}^2$ be the squared mutual distances, and let $K_{ij}$ denote the reduced radial symmetry operators associated with the $so(n-1)$ symmetry algebra.

Algebraicity conjecture. All symmetry operators $K_{ij}$ are algebraic differential operators in the variables $\rho_{ij}$, with coefficients linear in the $\rho_{ij}$.

The assertion concerns the explicit realization of the radial symmetry algebra in squared-distance coordinates. The supplied text gives no resolution status.