Local vertex-star basis conjecture for the Stokes potential space on Freudenthal meshes

Let $V_h\subset[H^1(\Omega)]^3$ be constructed with continuous Lagrange elements of degree $k$. Let $\Sigma_h\subset H^1(\operatorname{curl},\Omega)$ be the space preceding $V_h$ in a subcomplex

$$\mathbb{R}\xrightarrow{\operatorname{id}}S_h\xrightarrow{\operatorname{grad}}\Sigma_h\xrightarrow{\operatorname{curl}}V_h\xrightarrow{\operatorname{div}}\Pi_h\xrightarrow{\operatorname{null}}0.$$

A basis is supported on the stars of vertices when every basis function has support contained in the star of a vertex. Local vertex-star basis conjecture. For $k\geq5$, $\Sigma_h$ has a local basis supported on the stars of vertices on the Freudenthal meshes depicted in the source. Such a basis would support robust multigrid methods for the associated Stokes complex; the corresponding potential space is known on some special three-dimensional meshes, but its characterization on Freudenthal meshes remains open.