Bouquet-graph identity for topological recursion relations

Let $g$ be a non-negative integer. For $h\leq g$, let $\Gamma^\mathrm{loop}_{h,g-h,[n]}$ be the graph with one vertex of genus $h$ containing all markings and $g-h$ loops; these are called bouquet type graphs. Define the linear form $\mathcal F$ on polynomials in $\psi$-classes by

$$\mathcal F\left(\prod_{i=1}^n\psi_i^{k_i}\right)=\prod_{i=1}^n\frac{1}{(2k_i+1)!!}.$$

Bouquet-graph conjecture. For any topological recursion relation

$$\sum_\Gamma(\xi_\Gamma)_*(c_\Gamma)=0,$$

where each $c_\Gamma$ is a polynomial in $\psi$-classes, the bouquet-type coefficients satisfy

$$\sum_{h=0}^g8^{-h}\mathcal F\left(c_{\Gamma^\mathrm{loop}_{h,g-h,[n]}}\right)=0.$$

This is stated as a strengthening of an earlier conjecture and theorem concerning bouquet coefficients; the supplied text gives no resolution evidence.