  Let a contact 3-manifold $(Y, \xi_0)$ be the link of a normal surface
singularity equipped with its canonical contact structure $\xi_0$. We prove a
special property of such contact 3-manifolds of "algebraic" origin: the
Heegaard Floer invariant $c^+(\xi_0)\in HF^+(-Y)$ cannot lie in the image of
the $U$-action on $HF^+(-Y)$. It follows that Karakurt's "height of $U$-tower"
invariants are always 0 for canonical contact structures on singularity links,
which contrasts the fact that the height of $U$-tower can be arbitrary for
general fillable contact structures. Our proof uses the interplay between the
Heegaard Floer homology and N\'emethi's lattice cohomology.
