Uniqueness of the loop-graph coefficient in topological recursion relations

Let $g$ be a non-negative integer and consider a degree $g$ topological recursion relation of the shape

$$0=c\cdot\left(\xi_{\Gamma^\mathrm{loop}_{0,g,[n]}}\right)_*(1)+\cdots,$$

where $\Gamma^\mathrm{loop}_{0,g,[n]}$ is the stable graph with one genus-$0$ vertex equipped with $g$ loops and all the markings. Suppose that all omitted graphs have at least two vertices, that there are no $\kappa$ classes, and that no genus-$h$ vertex carries a monomial in $\psi$-classes of degree at least $h+\delta_h^0$ for $0\leq h\leq g$. Uniqueness conjecture. The coefficient $c$ must vanish. This is a uniqueness statement for topological recursion relations of the specified shape; the supplied text does not establish its resolution, so its status should be checked.