Locally conformally symplectic non-squeezing conjecture

Let $(\mathbb{R}^{2n} \times S^1 \times S^1,\omega)$ be the locally conformally symplectic manifold considered in the paper, and let $B_R$ be the radius-$R$ ball in $\mathbb{R}^{2n}$. Set

$$U:=B_R\times S^1\times S^1,$$

and let $\overline{U}$ denote its topological closure. A Hamiltonian lcs map is a locally conformally symplectic diffeomorphism generated by a smooth Hamiltonian function through the locally conformally symplectic Hamiltonian vector field.

Locally conformally symplectic non-squeezing conjecture. If $R\geq 1$, there is no compactly supported Hamiltonian lcs map

$$\phi:\mathbb{R}^{2n}\times S^1\times S^1\to\mathbb{R}^{2n}\times S^1\times S^1$$

such that $\phi(\overline{U})\subset U$.

This is proposed as a direct generalization of the contact non-squeezing theorem to the locally conformally symplectic setting. The supplied text does not state whether the conjecture has been proved or disproved.