Conjecture on transitory canard and saddle-node canard cycles

For $\varepsilon=0$, consider the transitory singular canard $\Gamma$, formed by the critical manifold in $[-1,1]$ and the segment

$$\{(x,k):x\in[-1,1]\}.$$

Here $\varepsilon>0$ is the perturbation parameter, $k$ is a system parameter, and the system is the one defined by the cited equations.

Transitory canard cycles conjecture. There exist values of $\varepsilon>0$ and $k\approx 1$ such that the corresponding system exhibits one, two, or three canard limit cycles in a neighborhood of $\Gamma$; moreover, it also exhibits zero, one, or two saddle-node canard cycles in a neighborhood of $\Gamma$.

This conjecture concerns the transition between three-zone and four-zone saddle-node canard orbits, which is not captured by the functions used in the paper. Its status is not resolved in the supplied source.