The comb-domain generic-position conjecture for KdV frequencies

Let $\eta=\{\eta_k\}_{k\geq1}$ and $\eta^{(1)}=\{\eta_k^{(1)}\}_{k\geq1}$ be the frequency collections associated with the comb domains describing the KdV flows. Comb-domain generic-position conjecture. With a fixed collection $\eta$, one can bring $\eta^{(1)}$ into a generic position by an arbitrarily small variation of $\{h_k\}_{k\geq1}$. This is a proposed step in the paper's program for constructing a counterexample; the supplied text gives no resolution.