The balanced separator conjecture for sphere intersection graphs

Let $t,d greater than or equal to 1$ be integers, and let $G$ be an $n$-vertex graph in the class $\SphereInt{d}_t$ of sphere intersection graphs. A balanced separator is a vertex set whose removal leaves no connected component with more than a fixed balanced fraction of the vertices. Balanced separator conjecture. Every such graph has a balanced separator of size

$${\bigO{}}_{t,d}(n^{1-\frac{1}{d+1}}).$$

The conjecture would improve the exponent in the known separator theorem for sphere intersection graphs. It is known for $d=1$, while the general case remains open.