No persistence along the slow curve after the focus-node bifurcation

Consider the system

$$\varepsilon^3\dot{x}=tx+y+\varepsilon^3c_1,\qquad \varepsilon^3\dot{y}=(t-b)x+ty+\varepsilon^3c_2.$$

Let $R$ denote the relief function and let $a$ and $b$ be the parameters appearing in the focus-node bifurcation, with $b$ the focus-node bifurcation point. The no-persistence conjecture. If a trajectory of this system goes along the slow curve in a neighborhood of a real $t$ with $t<a$ and $R(t)>R(b)$, then it does not go along the slow curve after the focus-node bifurcation point $b$. This is presented as a more precise conjecture motivated by a numerical simulation of the normal-form system; the paper does not prove it.