  In recent years a great deal of attention has been paid to discretizations of
the incompressible Stokes equations that exactly preserve the incompressibility
constraint. These are of substantial interest because these discretizations are
pressure-robust, i.e. the error estimates for the velocity do not depend on the
error in the pressure. Similar considerations arise in nearly incompressible
linear elastic solids. Conforming discretizations with this property are now
well understood in two dimensions, but remain poorly understood in three
dimensions. In this work we state two conjectures on this subject. The first is
that the Scott-Vogelius element pair is inf-sup stable on uniform meshes for
velocity degree $k \ge 4$; the best result available in the literature is for
$k \ge 6$. The second is that there exists a stable space decomposition of the
kernel of the divergence for $k \ge 5$. We present numerical evidence
supporting our conjectures.
