Coexistence conjecture for quasi-periodic motions and hyperbolic tori

In the outer, retrograde configuration, let $\mathcal{D}_{{{{{{{{{\rm t} t} t} t} t} t} t} t} s}$ be the $8$-dimensional phase-space region almost completely filled with a positive-measure set of $5$-dimensional KAM tori described in the preceding theorem. Coexistence conjecture. Full-dimensional quasi-periodic motions and hyperbolic $3$-dimensional tori coexist in $\mathcal{D}_{{{{{{{{{\rm t} t} t} t} t} t} t} t} s}$. This conjecture concerns the coexistence of stable quasi-periodic motions and hyperbolic motions in the same phase-space region; the source says it is likely to be proved somewhere, but gives no resolution.