Equality of the bifurcation parameters in the one-dimensional Liouville type equation

Let $(\lambda(\alpha),U(x;\alpha))$ and the constants $\alpha_1$, $\alpha_2$, and $\alpha_3$ be as in Theorem, where $\alpha_*<\alpha_1\leq\alpha_2\leq\alpha_3$ describe the degeneracy and bifurcation behavior of the solution branch. Equality conjecture. In Theorem,

$$\alpha_1=\alpha_2=\alpha_3.$$

The conjecture asserts that the first degeneracy, the non-even bifurcation, and the subsequent degeneracy occur at the same parameter value. Its resolution would sharpen the description of symmetry-breaking bifurcation for this one-dimensional Liouville type equation.