Generalized Wright conjecture for equations with negative Schwarzian

Consider the Wright delay differential equation

$$x'(t)=-x(t)+f(x(t-1)).$$

Assume that all conditions (H) on $f$ are satisfied. Generalized Wright conjecture. If $-f'(0)<\pi/2$, then

$$\lim\limits_{t\to +\infty}x(t)=0$$

for every solution $x(t)$. This generalizes Wright's conjecture in the setting of nonlinearities with negative Schwarzian derivative; the source states that the original Wright conjecture remains open, and proposes this formulation based on one-dimensional dynamics results for maps with negative Schwarzian derivative.