Conjectured dynamic convergence for the Hellan-Herrmann-Johnson Kirchhoff plate method

Let $e_w$ and $\bm{E}_{\kappa}$ denote the continuous Kirchhoff plate variables, and let $e_w^h$ and $\bm{E}_{\kappa}^h$ be their discrete approximations. The notation $a\lesssim b$ means $a\leq Cb$.

Kirchhoff plate convergence conjecture. Assuming a smooth solution of the Kirchhoff plate problem, the error estimates

$$\lVert e_w-e_w^h\rVert_{L^{\infty}(H^1)}\lesssim h^k, \qquad \lVert\bm{E}_{\kappa}-\bm{E}_{\kappa}^h\rVert_{L^{\infty}(L^2)}\lesssim h^k$$

should hold.

Optimal convergence of order $O(k)$ is known for the associated static problem under smoothness assumptions, but the corresponding dynamic estimates are only conjectured in the source.