Threshold classification conjecture for the normalized time-delay system

Consider the normalized delay differential equation referred to as, with solution $u(t;\beta,w_0)$ for parameter $\beta$ and initial velocity $w_0$. Let the comparison properties be those denoted by and. Threshold classification conjecture. For $\beta\geq 1$, there exists a unique value $g(\beta)\in(0,1)$ such that the solution $u(t;\beta,w_0)$ has the following behavior: if $w_0\in(g(\beta),1)$, there is a constant $T_*=T_*(\beta,w_0)>0$ satisfying

$$u(t;\beta,w_0)<1\quad(0\leq t<T_*),\qquad u(T_*;\beta,w_0)=1,\qquad u(t;\beta,w_0)>1\quad(t>T_*);$$

if $w_0\in(0,g(\beta)]$, then

$$u(t;\beta,w_0)<1\quad(t\geq0),\qquad u(t;\beta,w_0)\to1\quad\text{as }t\to\infty;$$

and the comparison properties and hold. The numerical experiments motivate this threshold description of the dynamics, but the existence and uniqueness of $g(\beta)$ and the stated comparison properties are conjectural.