Conjectured higher-order $L^2$ error estimate for the nonconforming hybrid method

Let $u$ be the exact solution of the two-dimensional vector Laplacian problem and $u_h$ its approximation by the order-$k$ nonconforming primal hybrid finite element method. Let $s$, $t$, and $\b5$ be the regularity and weighted-Sobolev parameters from the error estimates, and let $\b5_{\min}$ denote the minimum corner exponent. Under the conditions of the error-estimate theorem, except for the restriction $s\leq k$, higher-order $L^2$ error estimate conjecture.

$$\lVert u-u_h\rVert_\Omega \lesssim h^{\min(k+1,s+t)}\bigl(\lVert u\rVert_{s+1,1-\mu}+\lVert\nabla\cdot u\rVert_{s,\mu-1}+\lVert\nabla\times u\rVert_{s,\mu-1}\bigr).$$

The numerical experiments suggest that this estimate improves the previously established bound when $s>k$, while the source does not report a proof or disproof; its status therefore remains open.