The conjecture on asymptotics of rapidly oscillating solutions to delay equations

Let $s\in[1,2]$ be arbitrary. Consider the delay differential equation (1), with coefficient $c(t)$ and delay function $\Greekmath 011C (t)$ satisfying the assumptions in (10a), and suppose

$$\sup_{t\geq\Greekmath 011A }\int_{\Greekmath 011C (t)}^t|c(\Greekmath 0110 )|\,d\Greekmath 0110 =s,\qquad |c(\Greekmath 0110 )|\leq1,\qquad \forall\Greekmath 0110 \geq0,$$

where $\Greekmath 011A$ satisfies (10a). A solution is $\Lambda(s)$-rapidly oscillating according to the definition used in the source. Asymptotic-profile conjecture. All solutions of (1) that are $\Lambda(s)$-rapidly oscillating and nonnegative, and do not tend to zero, are asymptotic to $x_s$. The conjecture proposes a precise asymptotic profile for this class of non-decaying solutions; the source states that the problem remains open.