Conjecture on joined energy curves and bifurcation from zero and infinity

Let the curves in Figure be energy curves for the problem, with solutions parametrized by their energy and with parameter $\lambda\in\mathbb{R}$. The conjectured joined diagram is obtained by combining these curves into a figure similar to Figure. Joined energy-curve conjecture. For every $\lambda\in\mathbb{R}$, there exist two sequences of solutions, one with negative energy and the other with positive energy; consequently, bifurcation occurs from both $0$ and $\infty$. This would describe the possible complete energy curves and predict simultaneous bifurcation from the zero and infinite solution regimes; the source refers to further discussion in a later section, but provides no resolution here.