Convergence conjecture for nonresonant billiard normal forms

Let $E$ be the horizontal Euclidean space, let $I:E\to E$ be reflection in the horizontal hyperplane, and let $\hat{\beta}:\hat{\mathcal P}\to\hat{\mathcal P}$ be the quotient billiard map with fixed point $\hat{\gamma}$. Suppose its symplectic linearization at $\hat{\gamma}$ has eigenvalues $\lambda_1^{\pm1},\ldots,\lambda_n^{\pm1}$, where the set $\lambda_1,\ldots,\lambda_n$ is nonresonant, meaning that

$$\lambda_1^{k_1}\cdots\lambda_n^{k_n}\neq 1$$

for every nonzero $k=(k_1,\ldots,k_n)\in\mathbb Z^n$.

Convergence conjecture. If the eigenvalues additionally satisfy good Diophantine properties, then the formal Taylor series defining the function $f$ is locally convergent. The formal Taylor series defining the local conjugacy between $\hat{\beta}$ and its linearization is likewise locally convergent.

The question concerns analytic linearization of a locally integrable multidimensional billiard near a symmetric period-two orbit. The paper proves formal existence in the nonresonant case, while convergence under suitable Diophantine conditions is motivated by numerical evidence and remains open.