The variational solution threshold conjecture for the fractional Bernoulli problem on an interval

Let $b1 i (0,2)$, $x_0 i\tilde{\mathbb{R}}$, $r>0$, and $D=(x_0-r,x_0+r)$. Consider the variational problem referred to as Problem ref{variational_problem}, and the inner Bernoulli problem referred to as Problem ref{bernoulli_problem}, both on $D$.

Variational solution threshold conjecture. There exists a constant $bbda_{b1,D}$ with $bbda_{b1,D}>bbda_{b1,D}$ such that, for every $bbda a bbda_{b1,D}$, the variational problem has a unique solution; this solution is symmetric with respect to $x_0$, continuous on $\mathbb{R}$, nonincreasing on $[x_0,infty)$, and solves the inner Bernoulli problem. For every $bbda i (0,bbda_{b1,D})$, the variational problem has no solutions.

The conjecture compares the variational and nonvariational solution sets. It is motivated by the existence of at least two Bernoulli solutions for sufficiently large parameters and by classical results for balls, but the stated threshold and uniqueness remain unproved here.