The Hofer-distance conjecture for commutator-length strata

Let $(M,\omega)$ be a symplectic manifold, let ${\rm Diff}_{\rm Ham}(M)$ denote the Hamiltonian diffeomorphism group, and let ${\rm Diff}_{\rm Ham}^k(M)$ be the subspace of diffeomorphisms whose commutator length is at most $k$. For $\phi\in{\rm Diff}_{\rm Ham}(M)$, let $\|\phi\|_k$ be the associated seminorm defined using Hamiltonian fibrations over surfaces with at most $k$ handles. Hofer-distance conjecture. The distance, in the usual Hofer norm, between $\phi$ and ${\rm Diff}_{\rm Ham}^k(M)$ is equal to $\|\phi\|_k$.

This conjecture relates the fibration-based seminorms to the Hofer geometry of the commutator-length filtration. The source says that it is studied in the cited work, while the supplied text gives no resolution evidence.