Self-repulsiveness thresholds for regularized alpha-energy

Let $E_\alpha$ be the regularized $\alpha$-energy of a smooth closed surface. Self-repulsiveness with respect to $C^0$-topology means that sequences of embedded surfaces converging in $C^0$ to a singular surface with a double point cannot keep their energies bounded while distinct points approach one another in ambient distance. Also consider surfaces in $\mathbb{R}^3$ whose curvature is bounded by a constant $b>0$.

Self-repulsiveness conjecture. The regularized $\alpha$-energy $E_\alpha$ for smooth closed surfaces is self-repulsive with respect to $C^0$-topology if and only if $\alpha\leq -4$, whereas $E_\alpha$ restricted to the space of surfaces in $\mathbb{R}^3$ with curvature bounded by $b$ is self-repulsive if and only if $\alpha\leq -3$.

The local estimates preceding the conjecture show that orthogonal double points and cone singularities produce divergent energy for $\alpha\leq -4$, while tangential double points have the threshold $\alpha\leq -3$. The conjecture asserts that these thresholds characterize self-repulsiveness in the stated spaces; the supplied text gives no resolution.