The quasi-isometry conjecture for bounded sphere dimension

Let $d\geq 2$ be an integer, and consider the class of graphs with sphere dimension at most $d$. Quasi-isometry conjecture. This class is quasi-isometric to the class of intersection graphs of balls in $\mathbb{R}^d$.

If true, this would reduce the asymptotic-dimension question for graphs of bounded sphere dimension to ball intersection graphs. The reduction is known when $d=2$, but remains open for general $d\geq 2$.