Nonattainment conjecture for the one-dimensional Bianchi–Egnell quotient

Let $d=1$ and $s\in(0,\frac{1}{2})$. For $f\in\dot{H}^s(\mathbb{R})\setminus\mathcal B$, let $\mathcal Q(f)$ denote the Bianchi–Egnell quotient, let $c_{BE}(s)$ be its infimum, let $c_{BE}^{\text{loc}}(s)=\frac{4s}{d+2s+2}$ be the local limiting value, and let $\mathcal B$ be the manifold of Sobolev optimizers. Nonattainment conjecture. Then

$$c_{BE}(s)=c_{BE}^{\text{loc}}(s),$$

any minimizing sequence for $c_{BE}(s)$ converges to $\mathcal B$, and the Bianchi–Egnell inequality does not admit a minimizer. This predicts that the one-dimensional case differs from dimensions $d\geq 2$, where the quotient has values strictly below the local limiting value; the conjecture remains open.