Strong DR/DZ equivalence conjecture

Let $w^{\alpha}$ be the normal coordinates for the Dubrovin–Zhang hierarchy, let $\tilde u^{\alpha}$ be the normal coordinates for Buryak's double ramification hierarchy, let $\eta^{\alpha\mu}$ be the inverse metric, let $\bar h^{\mathrm{DZ}}_{\mu,0}$ be the corresponding Dubrovin–Zhang Hamiltonian, and let $\{\,\ \}_{K^{\mathrm{DZ}}}$ denote the Dubrovin–Zhang Poisson bracket. For any semi-simple cohomological field theory, there exists a differential polynomial $\mathcal{P}$ in the coordinates $w^{\alpha}$. Strong DR/DZ equivalence conjecture. The normal Miura transformation

$$w^\alpha \mapsto \tilde u^\alpha(w)=w^\alpha+\eta^{\alpha\mu}\partial_x\{\mathcal{P},\bar h^{\mathrm{DZ}}_{\mu,0}\}_{K^{\mathrm{DZ}}}$$

maps the Dubrovin–Zhang hierarchy to Buryak's double ramification hierarchy written in the normal coordinates. The explicit expression for $\mathcal{P}$ is given in the cited theorem; the conjecture concerns the claimed equivalence for every semi-simple cohomological field theory.