Conjecture on the monotonicity of Riemann sums for reciprocal quadratics

Let $f_b(x):=\frac{1}{1-bx+x^2}$, and let $\beta^+=\frac{3+\sqrt{13}}{4}$. Monotonicity conjecture.

1. $L_n(f_b)$ is neither increasing nor decreasing for all $b\in\big(\frac{1}{2},1\big)$ and is monotonically increasing for all $b\in(\beta^+,2]$.
2. $R_n(f_b)$ is neither increasing nor decreasing for all $b\in\big(1,\frac{3}{2}\big)$ and is monotonically decreasing for all $b\in(\beta^+,2)$.

The conjecture concerns the parameter ranges not covered by the preceding monotonicity theorem for left and right Riemann sums. The endpoint case $b=2$ in the first assertion is verified separately, while the remaining assertions are addressed by the paper's subsequent results.