Polynomial coercivity conjecture for the Ginzburg–Landau Hessian

Let $E”(u)$ be the Hessian of the Ginzburg–Landau energy at a minimizer $u$, and let $\rho(\kappa)^{-1}$ denote its coercivity constant. Assume the general hypotheses of the paper and let $\kappa\geq 1$. Polynomial coercivity conjecture. There exists $\alpha\geq 1$ such that

$$\rho(\kappa)^{-1}\lesssim \kappa^\alpha.$$

Equivalently, the coercivity scale satisfies $\rho(\kappa)\sim \kappa^\alpha$ according to the preceding numerical observation. No analytic dependence on $\kappa$ is known; the conjecture is intended to be verified numerically.