The multi-intersection problem for three or more sets

Let $(M,\omega)$ be a closed symplectic manifold, let $N\geq 3$, and let $D_1,\ldots,D_N$ be closed subsets of $M$. The multi-intersection problem asks whether there exist Hamiltonian diffeomorphisms $\varphi_1,\ldots,\varphi_N$ such that

$$\bigcap_i\varphi_i(D_i)=\varnothing.$$

Multi-intersection conjecture. Question has an affirmative answer provided there exist elements $\varphi_i$ lying in the identity component of the group of volume-preserving diffeomorphisms and satisfying the displayed empty-intersection condition. For $N\geq 3$, the paper describes the multi-intersection problem as likely soft, in contrast with the case $N=2$; the supplied text does not establish the asserted existence of such volume-preserving diffeomorphisms.