  In a recent proof of the log-concavity of genus polynomials of some families
of graphs, Gross et al. defined the weakly synchronicity relation between
log-concave sequences, and conjectured that the convolution operation by any
log-concave sequence preserves weakly synchronicity. We disprove it by
providing a counterexample. Furthermore, we find the so-called partial
synchronicity relation between log-concave sequences, which is (i) weaker than
the synchronicity, (ii) stronger than the weakly synchronicity, and (iii)
preserved by the convolution operation.
