Local hyperbolization of simple periodic solutions

Let $f \in \mathcal{C}^\infty(\Omega\times\mathbb{R}\times\mathbb{R}^d,\mathbb{R})$, and let $\mathcal{N}$ be any small open neighborhood of $f$ in $\mathcal{C}^r$. Let $p$ be a simple periodic solution of the scalar parabolic equation with minimal period $\omega>0$ such that

$$\sup_{t\in[0,\omega]}\|p(t)\|_{X^\alpha}\leq\tilde K,$$

where $\tilde K>0$.

Local hyperbolization conjecture. There exists a function $\tilde f\in\mathcal{N}$ such that $p$ is a hyperbolic periodic solution of the equation with nonlinearity $\tilde f$.

This statement is presented as the local result needed to prove the preceding density conjecture: it asserts that a given simple periodic orbit can be made hyperbolic by an arbitrarily small perturbation of the nonlinearity. Its resolution is not supplied in the source.