Distinct positive zeros of the component polynomials in the general case

Let $L$ be a nonnegative integer, let $c_0,ots,c_L$ be real numbers not all zero, and define

$$\Sigma(R)=\sum_{l=0}^L c_l D^l\left[x^{L+1}D^{L-l}(R)\right],$$

where $D^l$ denotes the $l$-th derivative. Set

$$R_1(x):=A_0(x):=\frac{x}{(1+x^2)^{3/2}},\qquad R_2(x):=B_0(x):=\frac{1}{(1+x^2)^{3/2}}.$$

Distinct-zeros conjecture. The positive zeros of $\Sigma(R_1)$ and $\Sigma(R_2)$ are distinct.

This is a polynomial formulation of the problem of distinguishing asymptotic directions of the zero sets of $X$ and $Y$ in the general configuration of point charges. The source gives no proof or resolution.