Generic mechanical non-volume-hyperbolicity conjecture

Let $f$ be a $C^r$ diffeomorphism, let $P$ be a saddle point, and let $\mathcal C(f,P)$ be its chain-recurrent class. The class is mechanically not volume-hyperbolic when its failure of volume-hyperbolicity is witnessed by mechanical failures of domination together with the corresponding unbounded volume sequences on the extremal bundles.

Generic mechanical non-volume-hyperbolicity conjecture. For any integer $r\geq 1$, there is a residual subset $\mathcal R$ of $C^r$-diffeomorphisms $f$ such that if $\mathcal C(f,P)$ is not volume-hyperbolic, then it is $C^r$-robustly mechanically not volume hyperbolic.

The conjecture would turn non-volume-hyperbolicity into a mechanism detectable robustly in the $C^r$ topology, and together with the paper's theorem would yield a characterization by accumulation of sinks or sources. The source gives no resolution.