Symmetry-breaking conjecture for the two-point variational problem

Let $q_{\mathrm{ic}}=\delta_{-\zeta}+\delta_{\zeta}$ for $\zeta>0$, set $\mathfrak{m}=1$ and $\xi_1=0$, and consider the variational problem in which $T$ and these data are fixed. Symmetry-breaking conjecture. There exists an $\alpha_{\mathrm{c}}\in\mathbb{R}$, depending on $T$ and $\zeta$, such that for every $\alpha_1>\alpha_{\mathrm{c}}$, the variational problem has exactly two minimizers, which are reflections of one another about $x=0$. This conjecture formalizes the predicted symmetry breaking for the minimizers and, if true, establishes nonuniqueness in this two-point setting; the supplied text gives no resolution.