Small-independent-set conjecture for triangle-free intersection graphs of boxes

For $\alpha>0$, an intersection graph of $n$ boxes in $\mathbb{R}^3$ has one vertex for each box and edges joining intersecting boxes; it is triangle-free when it has no $3$-cycle, and its independence number is the maximum size of a set of pairwise nonadjacent vertices. Small-independent-set conjecture for box graphs. For every $\alpha>0$ there exist $n$ and a triangle-free intersection graph of $n$ boxes in $\mathbb{R}^3$ with independence number at most $\alpha n$. Such a result would produce triangle-free box intersection graphs with independence number $o(n)$ and address the open question highlighted in the source for boxes in dimension three or higher.