Exponential closeness conjecture for the FitzHugh-Nagumo homoclinic C-curve

Let $C_{l,\epsilon}$ denote the left equilibrium of the FitzHugh-Nagumo system, let $q$ be the relevant saddle equilibrium, and let $W^u(C_{l,\epsilon})$ and $W^s(q)$ be their unstable and stable manifolds. Consider the region in parameter space where

$$W^u(C_{l,\epsilon})\cap W^s(q)$$

is nonempty. Exponential closeness conjecture. The C-curve of homoclinic bifurcations of the FitzHugh-Nagumo system turns exponentially close to the boundary of this region. The numerical calculations indicate that the sharp turn of the homoclinic C-curve occurs near the curve where the relevant invariant manifolds have a tangential intersection; the claimed exponential closeness remains unproved in the source.