Realization conjecture for the discrete Dirac nullity of triangulated spheres

Let an abstract triangulated sphere be realized in $\mathbb{R}^3$, and let $\operatorname{D}$ denote its discrete Dirac operator. A triangulated-sphere realization conjecture. Every abstract triangulated sphere admits a realization in $\mathbb{R}^3$ with

$$\dim\operatorname{Ker}\operatorname{D}=4,$$

so that it has no non-trivial infinitesimal conformal deformation with vanishing change in mean-curvature half-density. Consequently, for almost all triangulated spheres in $\mathbb{R}^3$, infinitesimal conformal deformations are exactly parametrized by functions $\rho:V\to\mathbb{R}$ satisfying

$$\sum_i\rho_i=0,$$

as the change in mean-curvature half-density. This would extend the type of rigidity conclusion associated with Gluck's theorem from the known convex-realization setting to spheres of general combinatorics.