Pseudo-injectivity radius conjecture for the symplectic two-sphere

Let $(S^2,\omega_{st})$ be the symplectic two-sphere normalized by

$$\int_{S^2}\omega_{st}=1.$$

Let $inj(M,\omega)$ be the pseudo-injectivity radius, namely the supremum of the radii $r$ such that $|\widetilde{\phi}|_j^s<r$ implies that the space of quasi-flat morphisms in $\mathcal{P}(id,\widetilde{\phi})$ is nonempty and contractible. For $\epsilon>0$, let $B(1-\epsilon)$ denote the Hofer $(1-\epsilon)$-ball in $\operatorname{Ham}(S^2,\omega_{st})$. Pseudo-injectivity radius conjecture.

$$inj(S^2,\omega_{st})=1.$$

Moreover, the space of quasi-flat morphisms from $id$ to $\phi$ over each $\phi\in B(1-\epsilon)$ forms a Serre fibration over $B(1-\epsilon)$. This conjecture concerns the local topology of quasi-flat morphisms near the identity in Hofer geometry; the source presents it as a further conjectural application, and no resolution is supplied.