Asymptotic completeness conjecture for edge-wave and surface-wave quasi-eigenvalues

Let $\{\tilde{\sigma}_j\}_{j\in\mathbb N_0}$ be the edge-wave and surface-wave quasi-eigenvalues arranged in ascending order. The sequence is asymptotically complete if the indexing function $k$ from the exponentially close correspondence with sloshing eigenvalues can be chosen so that there exist integers $N>0$ and $J\in\mathbb Z$ with

$$k(j)=j+J$$

for every $j>N$. Asymptotic completeness conjecture. The set of all edge-wave and surface-wave quasi-eigenvalues is asymptotically complete. This conjecture is introduced as the condition needed to obtain a quasi-eigenvalue next to each sloshing eigenvalue; the source does not report a proof or disproof.