Generic simplicity of eigenvalues along heteroclinic orbits

Let $\phi$ be a choice of nonlinearity in the semilinear parabolic equation referenced in

, and let $u$ be a heteroclinic orbit of that equation. For each time $t$, let $H(t)$ denote the linearized operator along $u$ and its eigenvalues. **Generic simplicity conjecture.** There is a generic \subset (a Baire \subset) of choices for $\phi$ in

such that, if $u$ is a heteroclinic orbit, all of the eigenvalues of $H(t)$ are simple.

This genericity condition is introduced to prevent eigenvalue bifurcations when constructing function spaces for the linearization about heteroclinic orbits. The supplied text does not state whether the claim has been proved or disproved.