The no-common-roots conjecture for the polynomials $W_r$ and $U_r$

Let $W_r(x)$ and $U_r(x)$ be the recursively defined polynomials associated with the unique exponential-Gaudin solutions indexed by $r\in\mathbb Z$. Polynomial-root conjecture. For every $r$, $W_r$ and $U_r$ have no multiple roots and no common roots; moreover, for $r\geq0$, $W_{-r}(x/2)$ and $U_{-r}(x/2)$ belong to $\mathbb Z_{\geq0}[x]$. The paper reports computer verification for small $|r|$ but gives no general proof.