  We show a counterexample to a conjecture of de Bobadilla, Luengo,
Melle-Hern\'{a}ndez and N\'{e}methi on rational cuspidal projective plane
curves. The counterexample is a tricuspidal curve of degree 8. On the other
hand, we show that if the number of cusps is at most 2, then the original
conjecture can be deduced from the recent results of Borodzik and Livingston
and (lattice cohomology) computations of N\'emethi and Rom\'an.
  Then we formulate a `simplified' (slightly weaker) version, more in the
spirit of the motivation of the original conjecture (comparing index type
numerical invariants), and we prove it for all currently known rational
cuspidal curves. We make all these identities and inequalities more transparent
in the language of lattice cohomology of surgery 3-manifolds $S^3_{-d}(K)$,
where $K$ is a connected sum of algebraic knots $\{K_i\}_i$.
  Finally, we prove that the zeroth lattice cohomology of this surgery manifold
depends only on the multiset of multiplicities occurring in the multiplicity
sequences describing the algebraic knots $K_i$. This result is closely related
to the lattice-cohomological reformulation of the above mentioned theorems and
conjectures, and provides new computational and comparison procedures.
