The asymptotic-dimension conjecture for bounded sphere dimension

Let $d\geq 1$ be an integer, and consider the class of graphs whose sphere dimension is at most $d$. Asymptotic-dimension conjecture. This class has asymptotic dimension at most $d$.

The bound would be tight because the hypercubic lattice $\mathbb{Z}^d$ has both asymptotic dimension and sphere dimension equal to $d$. The conjecture is known for $d=1$ and $d=2$, and an improved general upper bound of $d+1$ is known; the asserted bound $d$ remains open for general $d$.