Higher-order bounded isometry conjecture

Let $(M,\omega)$ be a symplectic manifold. For each integer $k\geq 0$, let ${\rm BI}_{k,0}(M)$ be the subgroup of $k$-bounded symplectomorphisms in $\operatorname{Symp}_0(M)$, and use the convention ${\rm BI}_{-1,0}(M)={\rm Ham}(M,\omega)$. Higher-order bounded isometry conjecture. For every integer $m\in{\mathbb N}\cup\\{0\\}$,

$$\bigcap_{k=m}^{\infty}{\rm BI}_{k,0}(M)={\rm BI}_{m-1,0}(M).$$

In particular, for every symplectic manifold $M$,

$$\bigcap_{k=0}^{\infty}{\rm BI}_{k,0}(M)={\rm Ham}(M,\omega).$$

This refines the bounded isometry conjecture by considering the hierarchy of metrics $d_k$ and their bounded isometry groups. The source presents it as a proposed generalization; no resolution is given.