Universality conjecture for inversion symmetry of WDVV tau functions

Let $F(v)$ be a solution of the WDVV equations, let $\hat F(\hat v)$ be its inversion, and let $\tilde{\mathcal F}_g$ and $\hat{\mathcal F}_g$ be the corresponding genus-$g$ free energies. Define

$$\mathcal G=\sum_{g\geq 1}\epsilon^{2g-2}\left(\tilde{\mathcal F}_g-\hat{\mathcal F}_g\right).$$

Let $\hat w^n$ be the associated transformed field. Universality conjecture. The difference can be represented as

$$\mathcal G=\mathcal G_1(\hat w^n)+\epsilon^2\mathcal G_2(\hat w^n)+\epsilon^4\mathcal G_3(\hat w^n)+\cdots,$$

where each $\mathcal G_k(\hat w^n)$ is a differential polynomial in $\hat w^n$, and the differential polynomials $\mathcal G_g$ do not depend on the particular solution $F(v)$ of the WDVV equations. The genus-one calculation preceding the conjecture gives the initial transformation law, while the higher-genus universal differential-polynomial representation is left open.