Dubrovin–Liu–Yang–Zhang conjecture on the standard form of tau-symmetric deformations

Let a tau-symmetric deformation of the Riemann hierarchy be given. A normal Miura transformation is a Miura transformation of the type defined in the source that preserves the relevant normalized form. Write $K_{\widetilde u}=\partial_x$ for the transformed operator, let $\mathcal P_{2g}'$ denote the source's set of partitions, and write $\widetilde u_\lambda$ for the corresponding jet monomials. Dubrovin–Liu–Yang–Zhang conjecture. There exists a unique normal Miura transformation $u\mapsto\widetilde u(u_*,\varepsilon)$ such that the transformed hierarchy is in standard form:

$$K_{\widetilde u}=\partial_x,$$ $$\overline h_1[\widetilde u]=\int\left(\frac{\widetilde u^3}{6}+\varepsilon^2a\widetilde u_x^2+\sum_{g\ge2}\varepsilon^{2g}\sum_{\lambda\in\mathcal P_{2g}'}a_\lambda\widetilde u_\lambda\right)dx,$$

where $a,a_\lambda\in\mathbb C$. If $a=0$, then $a_\lambda=0$ for all $\lambda$. If $a\ne0$, then all coefficients $a_\lambda$ are uniquely determined by $a$ and the coefficients $a_{(2^g)}$, $g\ge2$. This is a normal-form and classification claim for tau-symmetric deformations; the source gives no resolution.