I used 1 period. In that case, the dimensions of the static and the dynamic Jacobian are the same: 54. The rank is 52. For the static Jacobian I get a rank of 54. With three periods, the size is 162 and the dynamic Jacobian has rank 160.
In contrast, for the model with investment adjustment costs, I get all Jacobians to be full rank, which squares well with the solution.