Singular-limit convergence of C-shaped homoclinic bifurcation curves

Let $A=(p^*,0)$, $B=(p_-,0)$, and $C=(p_-,s^*)$, where $s^*\approx 1.50815$. Consider the C-shaped homoclinic bifurcation curves in $(p,s)$-parameter space for the FitzHugh–Nagumo equation with singular parameter $\epsilon$. C-shaped-curve convergence conjecture. As $\epsilon\to 0$, these curves converge to the union of the segments $AB$ and $AC$.

The conjecture is supported by numerical computations for $10^{-2}\geq\epsilon\geq 5\cdot 10^{-5}$ and by the preceding singular-limit analysis. The source does not specify a topology for the convergence or report a resolution.