The tau-function formulation of the Ablowitz–Ladik flow conjecture

Let $\tau$ be a tau function of the topological deformation of the Principal Hierarchy associated with the generalized Frobenius manifold defined in the paper, and let

$$\Lambda=\exp\left(\varepsilon\frac{\partial}{\partial x}\right).$$

Define the special two-point functions

$$U=\varepsilon(\Lambda-1)\frac{\partial\log\tau}{\partial t^{2,0}},\qquad W=(1-\Lambda^{-1})\left(\exp\left(\varepsilon\frac{\partial}{\partial t^{1,0}}\right)-1\right)\log\tau.$$

Tau-function Ablowitz–Ladik conjecture. The functions $U$ and $W$ satisfy the positive flows of the Ablowitz–Ladik hierarchy when $t^{2,k}$ is identified with $t_k$ for $k\geq 0$. The supplied text gives no resolution of this all-flow assertion.