Global stability conjecture for the equilibrium of the delayed quadratic-rational equation

Consider the delayed quadratic-rational difference equation referred to as Eq., with parameter $p$, and let $\bar{y}$ denote its equilibrium point. Global stability conjecture. Many numerical simulations suggest that if

$$\frac{1}{2}\leq p<\frac{3}{4},$$

then $\bar{y}$ is globally asymptotically stable. The preceding results establish global asymptotic stability for $0<p<\frac{1}{2}$ and local asymptotic stability for $0<p<\frac{3}{4}$; the stated parameter range is therefore the numerically supported extension of the global result, and remains unproved here.