Existence of a homoclinic bifurcation under predator harvesting

Consider the harvested predator-prey model given by Equation (HarvestingEqn), with all parameters fixed except the harvesting effort $q\geq 0$. A homoclinic loop is a trajectory that leaves a saddle equilibrium and returns to the same equilibrium. Homoclinic-bifurcation conjecture. There exists $q^*>0$ such that a homoclinic loop occurs when $q=q^*$. This conjecture describes a global bifurcation caused by predator harvesting; the supplied text gives no proof or resolution, so it remains open.