A lower bound for the ratio of the polynomials R_d and P_d

Let $R_d(x)$ and $P_d(x)$ be the polynomials whose ratio is under consideration. Lower-bound conjecture. If $d\ge5$ and $x\ge\frac12(d+\sqrt{d+2})$, then

$$-\frac{R_d(x)}{P_d(x)}\ge\frac{.995(d-2)}{2x+d-2}.$$

This conjecture is presented as information needed to make the paper's expression for the focal function $g(s_1,s_2)$ useful; the source provides no resolution evidence.