Quasi-flat minimizer conjecture for spectral length

Let $M$ be a symplectic manifold with symplectic form $\omega$, and let $\widetilde{\phi}\in\widetilde{\operatorname{Ham}}(M,\omega)$ be Floer non-degenerate. A generalized path from $id$ to $\widetilde{\phi}$ is an element of $\mathcal{P}(id,\widetilde{\phi})$, and $L^s_j$ denotes the signed spectral length functional. A morphism is quasi-flat when it realizes the lower bound $L^s_j(\mathcal{A})=|\widetilde{\phi}_+|_s-|\widetilde{\phi}_-|_s$. Quasi-flat minimizer conjecture. The functional $L^s_j$ attains a minimum on $\mathcal{P}(id,\widetilde{\phi})$. Moreover, this minimum can be represented by a quasi-flat morphism, and

$$|\widetilde{\phi}|_j=|\widetilde{\phi}|_s.$$

The conjecture asserts existence of geometrically distinguished minimizers for the spectral norm on the universal cover of the Hamiltonian symplectomorphism group; the stated theorem for Lalonde–McDuff Hamiltonian symplectomorphisms of $S^2$ provides a partial result, but the general claim remains open.