The boundedness conjecture for real finite-gap Boussinesq solutions

Let $u[K]$ denote the Abelian argument associated with a characteristic $[K]$, and let the finite-gap solution be the solution referred to in the source as the real finite-gap solution. The boundedness conjecture. If

$$\bm{C}=u[K],$$

then the finite-gap solution is bounded. This is a boundedness condition for the real finite-gap solutions constructed from the algebro-geometric data; the source does not state whether the conjecture has been proved or disproved.