Odd-trace vanishing versus individual theta-series vanishing

Determine whether vanishing of the odd trace of an odd unimodular lattice in a fixed degree implies that every odd-characteristic theta series of the lattice vanishes in that same degree.

Background

The paper studies odd-characteristic theta series attached to odd unimodular lattices and relates their signed sum, called the odd trace, to the difference between the theta series of the two even neighbors. In several rank-24 examples, the even neighbors have equal theta series in low degrees, so the odd trace vanishes. The authors explain that this does not automatically establish the vanishing of each individual odd-characteristic theta series. They explicitly leave unresolved whether vanishing of the aggregate odd trace forces all of its constituent series to vanish in the same degree.