Inf-sup stability conjecture for Scott–Vogelius elements on Freudenthal meshes

Let $V_h$ be constructed with continuous Lagrange elements of degree $k$, and choose $\Pi_h={{\nabla\cdot}\,}V_h$. The inf-sup condition is the existence of a constant $\beta>0$ such that

$$\beta\sqrt{\langle q,q\rangle}\leq\sup_{v\in V_h}\frac{\langle{{\nabla\cdot}\,}v,q\rangle}{\sqrt{\langle\nabla v,\nabla v\rangle}}$$

for all $q\in\Pi_h$. Inf-sup stability conjecture. For $k\geq4$, this condition holds on structured tetrahedral meshes of the Freudenthal type, with a constant that depends only on $k$. This conjecture concerns the stability of the Scott–Vogelius discretization of the three-dimensional Stokes problem; its validity on these Freudenthal mesh families is presented as an open question.