Generic multiplicity-two conjecture for Laplace eigenvalues of ellipses

Consider the family of ellipses

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$$

For each such ellipse, consider the multiplicities of the eigenvalues of its Laplacian. Generic multiplicity-two conjecture. For a generic class of ellipses, the multiplicity of each eigenvalue is at most $2$. The multiplicity question is largely open: for circular billiards, C. L. Siegel proved that the multiplicities are either $1$ or $2$, but the asserted generic bound for ellipses is presented as an open problem.