The $n^{-1}$ convergence-rate conjecture for deep-limit minimisers

Let $\theta^{(n)}$ be minimisers of the finite-depth energies $cE_n$, and suppose the assumptions of the convergence theorem for $cE_n$ and $cE_\infty$ are satisfied. Let $ctheta\in\Theta$ be a limit point of $\{\theta^{(n)}\}_{n\in\mathbb N}$, and let $\{\theta^{(n_k)}\}_{k\in\mathbb N}$ be a subsequence converging to $ctheta$. With $d$ denoting the distance on the parameter space, the conjecture asserts that there exists $C>0$ such that

$$d(\theta^{(n_k)},\theta)\leq C n_k^{-1}.$$

The $n^{-1}$ convergence-rate conjecture. Under these assumptions, the distance between the convergent subsequence of finite-depth minimisers and its limit is $O(n_k^{-1})$. This conjectured rate is motivated by Taylor expansions of the recovery sequence, which suggest an $O(n^{-1})$ rate for the minima when the relevant $H^2$ regularity is available, together with a local bound controlling parameter distance by the energy error. The result is not proved in the source.