Transversality conjecture for the first nullcline intersection

Let $\bar\gamma_{\bar\mu}$ be the special solution of the rescaled system, and consider its intersections with the nullcline $\bar y=\bar x^2$. Transversality conjecture. The least $t$ for which $\bar x'_{\bar\mu}=0$ corresponds to a transverse intersection. The authors state that only partial results are available.