The descent conjecture for the continuous Calabi quasi-morphism on the two-sphere

Let ${\mathcal P}^{ham}(Sympeo(S^2),id)$ denote the space of continuous Hamiltonian paths from the identity, let $\overline\mu^{path}$ be the extension of the homogeneous Calabi quasi-morphism, and let $\overline{ev}_1$ be the time-one evaluation map. Fathi's conjecture. The quasi-morphism descends to a homogeneous quasi-morphism $\overline\mu:Hameo(S^2)\to\mathbb R$ such that

$$\overline\mu^{path}=\overline\mu\circ\overline{ev}_1.$$

Equivalently, $\overline\mu^{path}(\lambda)$ depends only on the time-one map $\lambda(1)$ for $\lambda\in{\mathcal P}^{ham}(Sympeo(S^2),id)$. The conjecture is intended to address simplicity questions for area-preserving homeomorphisms; the corresponding smooth statement was proved by Entov and Polterovich, while the source gives no resolution for the continuous setting.