Let A, B, and C be three log-concave nonnegative sequences without internal zeros. Write AC and BC for their convolution sequences, and let A sim_w B mean that A and B are weakly synchronized. Gross et al.'s weak synchronicity conjecture. If A sim_w B, then AC and BC are weakly synchronized. The conjecture was proposed in connection with log-concavity criteria for genus polynomials and claims that weak synchronicity is preserved by convolution with any log-concave sequence. It is disproved by a counterexample in this paper.

Gross et al.'s weak synchronicity conjecture under convolution

Gross et al. proposed the conjecture in their work on log-concavity and genus distributions, treating it as open at the time. It asserts that weak synchronicity survives convolution with every log-concave sequence.

Known results

Gross et al., 2015: formulated the conjecture and connected it to log-concavity criteria for genus distributions.
The same work recorded no proof or counterexample.

January 2017 counterexample

A paper listed as published on January 1, 2017, reports that the conjecture is false and gives a counterexample. It introduces partial synchronicity, weaker than synchronicity but stronger than weak synchronicity, and states that this intermediate property is preserved under convolution by a log-concave sequence. No later proof, verification report, gap report, withdrawal, or retraction was found in the retrieved sources.

Current status (as of September 2026): The conjecture has a published report of a counterexample, but that resolution remains unverified in the retrieved record; no competing result was found.

Gross et al.'s weak synchronicity conjecture under convolution

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Progress summary

Solutions 0