Convolution Preserves Partial Synchronicity of Log-concave Sequences

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.
Comment: 13 pages