Generic diffeomorphisms are not robustly Turing-universal

Let $M$ be a compact computable manifold, possibly with boundary. Let $\mathcal{S}$ be a $C^\infty$ generic set of diffeomorphisms $f:M\to M$, and let $(f,\mathcal{E},\mathcal{D},\tau,\textnormal{\textsf{T}}_{\!\text{\rm univ}})$ denote a CDS extension of $f$.

Generic non-universality conjecture. For any compact computable manifold $M$ (possibly with boundary), there is a $C^\infty$ generic set $\mathcal{S}$ of diffeomorphisms $f:M\to M$ such that no diffeomorphism $f\in\mathcal{S}$ can be extended to a robustly Turing-universal CDS $(f,\mathcal{E},\mathcal{D},\tau,\textnormal{\textsf{T}}_{\!\text{\rm univ}})$.

This is a more precise formulation of the preceding generic non-universality expectation. It asserts the existence of a generic family of diffeomorphisms for which robust Turing universality is impossible; whether this holds for every compact computable manifold remains open.