Higher-codimension borderline minimality conjecture for elastic curves

Let $n\geq 2$ and let $\gamma\in W^{2,2}_{loc}(\mathbf{R};\mathbf{R}^n)$ satisfy

$$\inf_{\mathbf{R}}|\partial_x\gamma|>0.$$

A curve has a self-intersection if there exist distinct parameters with the same image point. Higher-codimension borderline minimality conjecture. If $\gamma$ has a self-intersection, then

$$E[\gamma]\geq 8,$$

with equality if and only if $\gamma$ coincides with the borderline elastica $\gamma_b$ up to orientation-preserving reparametrization and direction-preserving isometry. The planar case is established by the theorem preceding the conjecture; the claim is proposed for all codimensions, while its validity for $n\geq 3$ remains open.