Infinite-dimensional quasi-isometric geometry conjecture for the symplectic Banach–Mazur space of the sphere

Let $S^2$ be the two-dimensional sphere, let $\bar{\mathcal C}_{S^2}$ be the relevant completion or quotient space of persistence-module objects associated with $S^2$, and let $d_{SBM}$ denote its symplectic Banach–Mazur distance. A map is quasi-isometric if it distorts distances by multiplicative and additive constants. Sphere embedding conjecture. For every $N\in\mathbb{N}$, there exists a quasi-isometric embedding

$$\Phi:(\mathbb{R}^N,|\cdot|_{\infty})\to(\bar{\mathcal C}_{S^2},d_{SBM}).$$

The preceding results establish only one unbounded direction for $S^2$, whereas closed orientable surfaces of positive genus admit quasi-isometric embeddings of $\mathbb{R}^N$ for every $N\in\mathbb{N}$. The conjecture predicts that the sphere case likewise contains arbitrarily many unbounded directions, despite the contrast in fundamental groups.