A symmetric counterexample to Strang's conjecture for bivariate $C^1$ cubic splines on triangulations

We exhibit a triangulation of an equilateral triangle for which the space of bivariate $C^1$ cubic splines has dimension larger than the dimension formula conjectured by Strang. Notably, the triangulation is such that no two edges sharing a vertex are collinear, and the triangulation is invariant under the action of the isometry group ($D_3$) of the equilateral triangle which is triangulated.