Generic mechanical non-domination conjecture for chain-recurrence classes

Let $f$ be a $C^r$ diffeomorphism, let $P$ be a saddle, and let $\mathcal C(P,f)$ be its chain-recurrent class. The class is mechanically without domination of index $i$ when a cycle of basic sets containing $P$ contains a mechanical obstruction to domination of index $i$.

Generic mechanical non-domination conjecture. There is a residual subset $\mathcal R$ of $C^r$-diffeomorphisms $f$ such that if $\mathcal C(P,f)$ is not dominated of index $i$, then it has $C^r$-robustly mechanically no domination of index $i$.

This conjecture proposes that generic failures of dominated splittings arise from identifiable mechanisms such as non-simple Lyapunov spectra, tangencies, or bundle tangencies. The source gives no resolution.