Infinite-slope conjecture for the dispersal-induced growth transition curve

Assume $\bar{r}_1<0$, $\bar{r}_2<0$, and $\chi>0$, and let $\omega_{crit}(m)$ be the transition frequency for $m\in(0,m^*)$. The curve is understood to extend to the origin with $\omega_{crit}(0)=0$. Infinite-slope conjecture. The function $\omega_{crit}$ satisfies

$$\lim_{m\rightarrow 0}\omega_{crit}'(m)=+\infty.$$

Numerical results suggest that the transition curve is tangent to the $\omega$-axis at the origin, but the source states that this behavior has not been proved analytically.