Real separability conjecture for elliptic inflection polynomials

Let $f=x(x-1)(x-\lambda)$ be the Legendre cubic with three distinct real roots, so $\lambda\in\mathbb{R}\setminus\{0,1\}$. For each pair of indices $\mu,k$ considered in the paper, let $P_{\mu,k}(x,\lambda)$ be the corresponding inflection polynomial. Real separability conjecture. For every fixed real value of $\lambda$, $P_{\mu,k}$ has only simple roots in $x$ away from $\{0,1\}$. This would imply that each real zero $x=\gamma$ lifts to either two or zero real points of the inflectionary curve, according to the sign of $P_{\mu,k}(\gamma,\lambda)$.