Real-root count conjecture for elliptic inflection polynomials

Let $\mu\geq 1$ and $k\geq\mu+1$ be nonnegative integers, let $f$ be associated with a real Legendre parameter $\lambda$, and fix $\lambda\neq 0,1$. Let $P_{\mu,k}$ be the corresponding inflection polynomial. Real-root count conjecture. The polynomial $P_{\mu,k}$ has precisely either $\mu$ or $2\mu$ real roots $x=\gamma$ satisfying $f(\gamma,\lambda)>0$, according as $k-\mu$ is even or odd. The conjecture concerns the real locus of the inflectionary curve in the maximally real case and refines the preceding separability prediction by specifying the number of roots on the positive-real component.