Distinct positive zeros of the derivative-polynomial combinations

Let $A_0(x)=x(1+x^2)^{-3/2}$ and $B_0(x)=(1+x^2)^{-3/2}$, and let $A_l$ and $B_l$ be their $l$-th derivatives. Write

$$A_l(x)=\frac{P_l(x)}{(1+x^2)^{(2l+3)/2}},\qquad B_l(x)=\frac{Q_l(x)}{(1+x^2)^{(2l+3)/2}},$$

where $P_l$ and $Q_l$ are the corresponding polynomials. Suppose that $\gamma_l$ are real numbers which are not all zero.

Polynomial distinct-zeros conjecture. The positive real zeros of the polynomials

$$\sum_{l=0}^L\gamma_l(1+x^2)^{L-l}x^lP_l(x)$$

and

$$\sum_{l=0}^L\gamma_l(1+x^2)^{L-l}x^lQ_l(x)$$

are distinct.

This is presented as another version of the preceding conjecture. The source notes that interlacing and universal reality of the zeros fail in examples, but distinctness is conjectured and remains unresolved.