Dubrovin–Zhang hierarchy conjecture for homogeneous CohFTs

Let a homogeneous cohomological field theory be given, with potential $\mathcal F(t^*_*,\varepsilon)$, formal variables $t^\alpha_a$, and topological variables $w^{\mathrm{top};\alpha}_n$. Let $\widehat{\mathcal A}_w$ be the ring of differential polynomials in $w^1,\ldots,w^N$, and let $\Omega_{\alpha,a;\beta,b}$ and $\overline h_{\alpha,d}$ be as in the assertions below. Let $\eta$ be the metric, $\mu_\alpha=q_\alpha-\frac{\delta}{2}$, and $A^\alpha_\beta=\eta^{\alpha\nu}A_{\nu\beta}$.

Dubrovin–Zhang hierarchy conjecture. (1) For every $1\leq\alpha,\beta\leq N$ and $a,b\geq0$, there is $\Omega_{\alpha,a;\beta,b}\in\widehat{\mathcal A}_{w;0}$ such that

$$\frac{\partial^2\mathcal F}{\partial t^\alpha_a\partial t^\beta_b}=\left.\Omega_{\alpha,a;\beta,b}\right|_{w^\gamma_n=w^{\mathrm{top};\gamma}_n}.$$

(2) There is a Poisson operator $K_1^{\mathrm{DZ}}$ for which $\overline h_{\alpha,-1}:=\int\eta_{\alpha\nu}w^\nu\,dx$ are Casimirs and

$$\eta^{\alpha\mu}\partial_x\Omega_{\mu,0;\beta,b}=K_1^{\mathrm{DZ};\alpha\nu}\frac{\delta\overline h_{\beta,b}}{\delta w^\nu},$$

where $\overline h_{\beta,b}:=\int\Omega_{\boldsymbol{1},0;\beta,b+1}\,dx$. (3) There is a Poisson operator $K_2^{\mathrm{DZ}}$ such that

$$\{\cdot,\overline h_{\alpha,d}\}_{K_2^{\mathrm{DZ}}}=\left(d+\frac{3}{2}+\mu_\alpha\right)\{\cdot,\overline h_{\alpha,d+1}\}_{K_1^{\mathrm{DZ}}}+A^\beta_\alpha\{\cdot,\overline h_{\beta,d}\}_{K_1^{\mathrm{DZ}}},\qquad 1\leq\alpha\leq N,\ d\geq-1.$$

Parts (1) and (2) are known for arbitrary semisimple CohFTs, while the full homogeneous conjecture, including the second Poisson structure and recursion, remains open in the source.