Tau-function representation conjecture for vessel scattering matrices

Let $S(\lambda,x)$ be the vessel matrix and let $\tau(x)$ be the associated tau-function. The entries of $S(\lambda,x)$ are understood with respect to some $p$-norm on $\mathrm I$. Tau-function representation conjecture. The entries of the matrix $S(\lambda,x)$ are linear combinations of $\dfrac{\tau^{(n)}(x)}{\tau(x)}$ in some $p$-norm on $\mathrm I$; equivalently, each entry has the form

$$\sum \alpha_n \dfrac{\tau^{(n)}(x)}{\tau(x)}, \quad \alpha_n\in\mathbb C.$$

This conjecture is presented as a generalization of the finite-dimensional result cited by the authors. It concerns representing the vessel matrix, and consequently the potential and solutions of the output linear differential equation, in terms of the tau-function; its resolution is not specified in the source.